Dumb math question...

[bluejay]

What's the correct way to write out multiplication tables?  If I want to make a table of "times two's" do I make it like this?

 

1 x 2

2 x 2

3 x 2

 

... and so on.  Or like this?

 

2 x 1

2 x 2

2 x 3

 

... and so on?

 

 

In other words, which one is "two groups of one" and which one is "one group of two"?  I've been looking at math table posters for sale and also printable math tables.  Some do it one way, some do it the other way. 

 

Edited June 1, 2017 by bluejay

[wapiti]

https://beastacademy.com/pdf/3B/printables/BlankTimesTable.pdf

 

 

https://beastacademy.com/pdf/3B/printables/MixedTimesTable.pdf

 

In other words, which one is "two groups of one" and which one is "one group of two"? 

 

It doesn't matter.

[Farrar]

https://beastacademy.com/pdf/3B/printables/BlankTimesTable.pdf

 

 

https://beastacademy.com/pdf/3B/printables/MixedTimesTable.pdf

 

 

It doesn't matter.

 

I agree. Except... it does to some standardized tests and some classroom teachers.

 

https://blogs.scientificamerican.com/roots-of-unity/teaching-the-controversy-is-5-3-five-3s-or-three-5s/

[bluejay]

Thank you.  I know the answer is the same, and in that sense it doesn't matter.  But WILL it matter down the road?  I always thought of 5x3 and 3x5 as three groups of five either way.  It is simple when memorizing math facts and tables.  But wouldn't it matter when you demonstrate with manipulatives?  And as Farrar says, with standardized tests?  It is kinda confusing.

[Farrar]

Obviously if you're, say, getting some boxes of crayons for individual kids to use in a school, it's way better to get 1000 boxes of 12 crayons (1000 x 12) and not 12 boxes of 1000 (12 x 1000). So, yeah, I guess in the real world then the notation matters a little. But I wouldn't personally mark a story problem that solved the number of crayons as being 12 x 1000 as wrong. Some schools seem to have a different take, but I think the Scientific American thing I posted covers better than I can say why I think that approach is all wrong.

[Hunter]

I think students need to see them both ways.

 

I am noticing a concerning trend of the new tests requiring students to memorize rigid and maybe even wrong rules and definitions for very basic things that are  taught "wrongly" in texts other than those aligned with the test.

 

Often free vintage texts are considered "wrong". What are these tests going to do to any text older than a couple years old? Now 2 year old texts are "wrong". The literary analysis essay lesson put up by Time Magazine for Kids wrongly teaches how to write about theme, I guess. According to the new tests, theme cannot be a one-word topic like friendship. 

 

I find this trend concerning. I wonder what the colleges think about students arriving with rigid ideas that don't match the college textbooks and the lessons the professors have been teaching their entire careers. I can't see the colleges adopting this silliness. Maybe some won't even tolerate it, and in some instances mark the students wrong the opposite way?

 

What will the workforce think about these students? What will Time Magazine think about interns arriving with ideas that their website lessons are wrong?

 

I'm really slow to believe a text is "wrong" now, just because I'm told that it is.

 

[mamakelly]

I was taught that the x symbol means "groups of".

 

So 1x2 =one group of 2

And 2x1 = two groups of 1

[scoutingmom]

Some countries do it one way, some do it the other. In North America, some curriculum do it one way, some do it the other. It makes no real difference to the answer, and if for some reason you need (from the above example) to find out how many boxes of something an equation means, you would need to use common sense or ask. But then, I have never in real life seen someone say something like "buy 8x100 widgets" or "buy 100x8" widgets... usually there are words involved.

 

I had a math teacher in one grade that required a particular order. I had one another year that required a different order. Most didn't care. I learned to just do it what way the particular teacher wanted. No idea what to suggest for standardized tests though.

 

 

 

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[mathmarm]

What's the correct way to write out multiplication tables?  If I want to make a table of "times two's" do I make it like this?

 

1 x 2<-- 1 group   of 2

2 x 2<-- 2 groups of 2

3 x 2<-- 3 groups of 2

 

... and so on.  Or like this?

 

2 x 1 <--2 groups of 1

2 x 2 <--2 groups of  2

2 x 3 <--2 groups of  3

 

... and so on?

 

 

In other words, which one is "two groups of one" and which one is "one group of two"?  I've been looking at math table posters for sale and also printable math tables.  Some do it one way, some do it the other way. Because the makers of those posters don't know or care that they may have different meanings. Most math products manufactured in this country are just that--products they aren't intended to showcase mathematical meaning or reinforce correct teaching. Half the time, even when the products are accurate, the creators don't even know it.

To the best of my understanding this isn't a dumb question. Arithmetically speaking 1 x 2 and 2 x 1 have different meanings. They are equivalent, in that they give the same value, but they are different.

 

Realistically, "2" is the final answer to an infinite number of mathematical problems, but that doesn't make all of those math problems the same.

 

Consider 1+1, |-2|, 18-16, 2²-2, -(-2) and d/dx(2x). In each case, the "answer" is 2. But they all mean different things.

 

In the expression 1 x 2 1 is the multiplier and 2 is the multiplicand. Where as,

In the expression 2 x 1 2 is the multiplier and 1 is the multiplicand.

 

Yes, they give the same result, and yes, multiplication is commutative, but language use can erode the meaning or even the mention of a word. But what language use hasn't done was erode the meaning of the mathematical statement itself.

 

The properties of real numbers come from the abstraction. Whether I have 6 marbles in 4 bags, or  4 marbles in 6 bags, it's true that I'll have the same number of marbles in the end. BUT those two situations are different.

 

Because in the US we've had generations on generations of poorly written math texts and mathematically incompetent teachers we'll probably never eradicate the sloppy mis-use of our Human language (in this case English) in mathematics.

 

So long as we continue to ignore the existence of those words that make arithmetic precise and give each individual number a distinct role, then try, as they might, test makers will never get children (or parents for that matter) to give the answer that they are expecting.

 

 

There are many palindromes in English (words that are the same whether read left-to-right, or right-to-left: mom, dad, bob, sis, level, etc) but they are all meant to be read left-to-right. 

 

Just cause a beginning reader does it backwards and gets the word anyway, doesn't mean that they aren't making a mistake. 

[Farrar]

Just cause a beginning reader does it backwards and gets the word anyway, doesn't mean that they aren't making a mistake. 

 

I guess the reason I take exception to this to some extent is that it's purely notational and - as pointed out above - there are textbooks that disagree and different countries that disagree. And they can, because we could have decided that it read the other way. It's like periods vs. commas in American vs. British math. Or where to put the commas in American vs. Indian math. It's nothing to do with the math itself. It's just about the notation.

 

I'm not saying it's not something to teach. But I think it's radically different from a new reader reading the words all wrong, which does change their meaning. I would worry that if we strongly emphasize the idea of "wrong" with a student who is reading 2 x 3 as three groups of two or showing their array on it's "side", we obscure the much more important idea that multiplication is commutative.

[justasque]

I always thought of 5x3 and 3x5 as three groups of five either way. It is simple when memorizing math facts and tables. But wouldn't it matter when you demonstrate with manipulatives? And as Farrar says, with standardized tests? It is kinda confusing.

With manipulatives, you make a rectangle. Kid makes three groups of five. That can then be made into a rectangle with the groups horizontal or vertical. Doesn't matter which so long as the kid grasps that the rectangle is still a representation of three groups of five. Kid can then count the items to get fifteen. Parent can, if it is a good teachable moment, point out variations and/or relationships by skip counting, rearranging the items, etc. ("Hey, check this out, we can take the same rectangle and make it five groups of three! That's because multiplication is commutative - you can rearrange the numbers and the answer is the same. Neat!")

 

I have never seen a standardized test question that required a student to arrange in a certain way, or notate in a certain way. But there are questions designed to get at the understanding that multiplication is commutative.

 

 

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[mathmarm]

Thank you.  I know the answer is the same, and in that sense it doesn't matter.  But WILL it matter down the road?  I always thought of 5x3 and 3x5 as three groups of five either way.  This would be where you are wrong.  5x3 and 3x5 are fifteen either way, but they're not three groups of five, either way.

 

It is simple when memorizing math facts and tables. 

But wouldn't it matter when you demonstrate with manipulatives? Bearing in mind that I observe--and encourage everyone to observe--the differences between a multiplier and multiplicand, yes I think it matters when you demonstrate with manipulatives.

 

A x B, IN ENGLISH is supposed to mean A # of groups, with B # of item in each.  Or said more straight forward, A groups of B size.

 

Think of it as arranging chairs in an auditorium, or putting grapes into snack bags, or collecting money from various people.

 

The array representation as area of a rectangle is ONE way to think of multiplication, and it's limited.

 

It's my preference and my professional recommendation that you teach the difference between 2x3 and 3x2, because there is one.

 

It's helpful to start with an amount of counters and arrange them in to different groups. The amount doesn't change, but the arrangement sure does. 

And as Farrar says, with standardized tests?  It is kinda confusing.

I'm not sure about the standardized tests, but I can guarantee that I'd rather my children be mathematically precise than score well on a test written and approved by the mathematically incompetent.

 

Edited June 2, 2017 by mathmarm

[Hunter]

Sometimes precision doesn't result in ONE answer. Neither color or colour are wrong. Neither gray or grey is wrong. There are plenty of tests that require one or the other spelling, but it is a dumb test that specifically shows a child the spellings of both countries and only marks their own country's spelling correct.

 

 

[mathmarm]

Except mathematics isn't spelling in a human language. So the grey-gray and colour vs color thing aren't a good analogy. Spelling is strictly a human convention and has no roots to the physical world. The mathematical expressions in question model one real-world situation but not another. Spelling does not.

 

Yu kan reed ths jst fyn.

Soh "color" vursis "colour" izent a gould eksampul.

Eevin win eye ryt "kolur" Yoo kan reed it buht th spehlng iz knot ryt.

 

My point is, that spelling doesn't model any real world phenomena. Mathematics does. Or it's meant to anyway.

 

The OP asked specifically about 1x2 vs 2x1 for teaching purposes with manipulatives. She can teach it incorrectly or correctly no matter what, but since she's asking in how best to teach it and trying to understand WHY it should be one way or the other, then it's really not helpful to make arguments for why she should just do it anyway or "wrong".

 

I'm sure that there are engineers and doctors out there who don't know the difference and learned it without understanding that, so it's like missing this point in elementary math will ruin a person, but there is a distinction. Since the OP is specifically seeking to understand if there is a distinction and WHAT the distinction is, I think it's important that we not let our personal experiences/feelings get in the way of saying 

 

"Yes, OP, there is a difference, and the difference is ____"

 

As we all already know, 1 x 2 and 2 x 1 are both expressions for 2, but to answer the OPs question they DO have different meanings and when teaching multiplication as a concept, it'd be more correct to explicitly teach that they do, in fact, model different situations.

 

This isn't about the orientation of one rectangle vs another. To me understanding the commutitive property of multiplication speaks more to the fluidity of a single quantity than the orientation of a rectangle.

 

How many different ways can this quantity be arranged into various equally-sized groups?

 

Is it easier to carry 32 things in 4 boxes with 8 things in each box, or 8 boxes with 4 things in each? 

[winterbaby]

My understanding is that "x times y" means "y, taken x times," in other words x groups of y. But I think to a lot of people, especially kids, "x times y" feels like a subject-verb-modifier sentence so they think you take the number x and - kids will often say precisely this - "times it," i.e. take it y number of times. That's the logic of the saying "any number times one is itself," which seems to come from the educational establishment. I don't think this is terribly crucial but when my daughter, who must have learned it in school, started answering "x times one" as "itself!" I made sure to correct her. I mean, whichever way is right or wrong or whether it matters, answering a math problem with a pronoun is unacceptably ambiguous anyway. I think we would need another way of talking about this altogether to make it at all clear to most people.

[MrSmith]

I guess the reason I take exception to this to some extent is that it's purely notational and - as pointed out above - there are textbooks that disagree and different countries that disagree. And they can, because we could have decided that it read the other way. It's like periods vs. commas in American vs. British math. Or where to put the commas in American vs. Indian math. It's nothing to do with the math itself. It's just about the notation.

 

I'm not saying it's not something to teach. But I think it's radically different from a new reader reading the words all wrong, which does change their meaning. I would worry that if we strongly emphasize the idea of "wrong" with a student who is reading 2 x 3 as three groups of two or showing their array on it's "side", we obscure the much more important idea that multiplication is commutative.

[i lied.]

 

I think we can all agree that 2x3=3x2 (in the set of reals), but if I bring two sets of 3 balls to juggle with that third kid will be watching the other two.

Edited June 2, 2017 by MrSmith

[Hunter]

I don't believe that phonics and mathematics are all that different. Different countries notate math differently.

 

ExH made up his own math notation. I had to make sure the boys did not copy him. It is so stuck in my head how much math can be done with "wrong" notation. Helicopters can be fixed. 2-3 story high Japanese newspaper recycling machines can be built. Prototypes for surgical equipment can be developed. And on and on. Sure he mostly had to work alone, because coworkers were tearing their hair out. The head welder didn't know what was going on, but just did his job and let ExH do his.

 

There is a lot of math that is beyond me. But seeing math of other countries, watching my son do math with Roman numerals and Greek letters and the methods that must be used when using that type of notation, and watching ExH fight with his coworkers, I've been exposed to more than someone as low functioning as I am is typically exposed to.

 

My whole life I've been jumping from culture to culture hearing one culture scream the other is "wrong". I'm all wronged out. The whole "theme vs topic" debate is when I just stopped automatically believing less rigid definitions are wrong or incomplete, even in subjects I know only enough to know how much I don't know.

 

 

 

 

 

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[winterbaby]

But the difference between one group of two and two groups of one is not notational. They are similar in one sense - the total number of objects - but the sense in which they are different exists independent of notation. Arguably our notation is inadequate, if it obscures the fact that there is a difference at all from so many people. But that's not an argument for those who do know to obscure it deliberately, or to claim that the difference isn't there in reality.

[wapiti]

Here is an old article on this issue, arguing that it matters.  https://medium.com/i-math/why-5-x-3-5-5-5-was-marked-wrong-b34607a5b74c  I agree with the comments to the article that say no, the order still doesn't matter.  To quote one of them:

 

For what it’s worth I’m a former math teacher and I appreciate Brett’s effort to encourage trust in good teaching. But this is not good teaching and Brett is wrong to defend it. While, as the Wikipedia article points out, three groups of four is “often written as 3 x 4â€, this is not equivalent to a definition of the meaning of “3 x 4â€. Yes, “3 x 4" can mean three groups of four, but it can also mean four groups of three. In situations where the distinction is important, other means than syntax must be employed to indicate the difference. There simply is no standard, widely accepted interpretation of the syntax of multiplication. I suspect that a poll of mathematicians would reveal as much. For a teacher to declare that their interpretation is the standard, and then insist that students follow it is just bad teaching. Yes, syntax is important in programming and in matrix multiplication. No, insisting that students follow a made-up syntax for multiplication of whole numbers is not a good way to teach this.

 

Being an effective math teacher is not easy. It requires trying to understand the ways that a student interprets a concept and then helping the student to expand their thinking to include other interpretations. This is not accomplished by enforcing non-existent rules of notation.

 

What might help me understand the position that order matters would be the presentation of an actual elementary math problem where the order matters.

Edited June 3, 2017 by wapiti

[Kiara.I]

A x B, IN ENGLISH is supposed to mean A # of groups, with B # of item in each.  Or said more straight forward, A groups of B size.

 

But, equally...

It also gets argued the other way.

 

2+3 is 2, and then three is added to it.

2-3 is 2, and then three is subtracted from it.

 

So the first number in the sequence is what you start with, and then the second is the size of the action performed upon it.

 

Which to be consistent would lead to:

 

2 x 3 is  2, taken three times.  So, three groups, each of size 2.  B groups, of A size.

[bluejay]

 

 

Thank you, everyone.  And thanks for the explanations.  Yes, growing up, I always visualized 3x5 and 5x3 as three groups of five.  I always imagined it was a [smaller number] times [larger number] because it was quicker and easier.  I wasn't concerned about getting the grouping right.  No one asked. :D

 

However when homeschooling, this is not the way I want to teach.  I asked the question because I want to make tables, a "times one" table, a "times two" table and so on.  I just didn't know if it should be 1 x 2 or 2 x 1 because these "sources" were all doing it differently.  I'm inclined to write a "times one" table as "2 x 1" ("two groups of one") and so on, but I wanted to ask for your advice.  :)  I will admit I'm not a math-obsessed person and it would've been fine with me any way.  This isn't about my own preferences though, but about doing it right.

 

On consulting the Saxon text, it says indeed 2 x 5 is "two groups of five" and 5 x 2 is "five groups of two."  Oddly for me, Saxon spells out 2 x 5 as    5   and says that is "two groups of five."  

                                          x2

                                         ----

 

What?  I'm kinda used to reading problems like this from top to bottom, so I'd read that as "five times two."  Of course, Saxon acknowledges "switcharound facts."  Maybe I will find a clarification when I read the whole text!   Still thinking of switching to Singapore Math though.

[scoutingmom]

To the best of my understanding this isn't a dumb question. Arithmetically speaking 1 x 2 and 2 x 1 have different meanings. They are equivalent, in that they give the same value, but they are different.

 

Realistically, "2" is the final answer to an infinite number of mathematical problems, but that doesn't make all of those math problems the same.

 

Consider 1+1, |-2|, 18-16, 2²-2, -(-2) and d/dx(2x). In each case, the "answer" is 2. But they all mean different things.

 

In the expression 1 x 2 1 is the multiplier and 2 is the multiplicand. Where as,

In the expression 2 x 1 2 is the multiplier and 1 is the multiplicand.

 

Yes, they give the same result, and yes, multiplication is commutative, but language use can erode the meaning or even the mention of a word. But what language use hasn't done was erode the meaning of the mathematical statement itself.

 

The properties of real numbers come from the abstraction. Whether I have 6 marbles in 4 bags, or 4 marbles in 6 bags, it's true that I'll have the same number of marbles in the end. BUT those two situations are different.

 

Because in the US we've had generations on generations of poorly written math texts and mathematically incompetent teachers we'll probably never eradicate the sloppy mis-use of our Human language (in this case English) in mathematics.

 

So long as we continue to ignore the existence of those words that make arithmetic precise and give each individual number a distinct role, then try, as they might, test makers will never get children (or parents for that matter) to give the answer that they are expecting.

 

 

There are many palindromes in English (words that are the same whether read left-to-right, or right-to-left: mom, dad, bob, sis, level, etc) but they are all meant to be read left-to-right.

 

Just cause a beginning reader does it backwards and gets the word anyway, doesn't mean that they aren't making a mistake.

But it isn't standard which is which worldwide or even in North America. So it really doesn't matter the order because it is not agreed on.

 

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[scoutingmom]

Except mathematics isn't spelling in a human language. So the grey-gray and colour vs color thing aren't a good analogy. Spelling is strictly a human convention and has no roots to the physical world. The mathematical expressions in question model one real-world situation but not another. Spelling does not.

 

Yu kan reed ths jst fyn.

Soh "color" vursis "colour" izent a gould eksampul.

Eevin win eye ryt "kolur" Yoo kan reed it buht th spehlng iz knot ryt.

 

My point is, that spelling doesn't model any real world phenomena. Mathematics does. Or it's meant to anyway.

 

The OP asked specifically about 1x2 vs 2x1 for teaching purposes with manipulatives. She can teach it incorrectly or correctly no matter what, but since she's asking in how best to teach it and trying to understand WHY it should be one way or the other, then it's really not helpful to make arguments for why she should just do it anyway or "wrong".

 

I'm sure that there are engineers and doctors out there who don't know the difference and learned it without understanding that, so it's like missing this point in elementary math will ruin a person, but there is a distinction. Since the OP is specifically seeking to understand if there is a distinction and WHAT the distinction is, I think it's important that we not let our personal experiences/feelings get in the way of saying

 

"Yes, OP, there is a difference, and the difference is ____"

 

As we all already know, 1 x 2 and 2 x 1 are both expressions for 2, but to answer the OPs question they DO have different meanings and when teaching multiplication as a concept, it'd be more correct to explicitly teach that they do, in fact, model different situations.

 

This isn't about the orientation of one rectangle vs another. To me understanding the commutitive property of multiplication speaks more to the fluidity of a single quantity than the orientation of a rectangle.

 

How many different ways can this quantity be arranged into various equally-sized groups?

 

Is it easier to carry 32 things in 4 boxes with 8 things in each box, or 8 boxes with 4 things in each?

But a lot of things in math notation is a made up convention. And this convention isn't standardized. The way you say is 'right', is considered wrong in some other countries. Or with a different math course. Because it isn't standard, either way is right. Or either way is wrong.

 

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[scoutingmom]

But the difference between one group of two and two groups of one is not notational. They are similar in one sense - the total number of objects - but the sense in which they are different exists independent of notation. Arguably our notation is inadequate, if it obscures the fact that there is a difference at all from so many people. But that's not an argument for those who do know to obscure it deliberately, or to claim that the difference isn't there in reality.

The notation isn't standardized, no matter what your elementary math teacher said!

 

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[scoutingmom]

http://mathforum.org/library/drmath/view/58567.html

 

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[happypamama]

I would read 2x1 as "two groups of one," meaning that if I'm giving cookies to both little boys, I will give one cookie to each. The other way around would mean I give both cookies to one boy.

 

But I think Singapore actually reads it the other way, as in 2x1 would mean 2, taken 1 time. And whereas I would write a times table as 1x2, 2x2, 3x2, etc., meaning "one group of two, two groups of two, three groups of two, etc.," I think they would write it as 2x1, 2x2, 2x3, etc., meaning "two taken one time, two taken two times, two taken three times, etc."

[Farrar]

I guess what it boils down to for me is that it's important that kids understand that 2 groups of 3 is different from 3 groups of 2 in the real world. But that those things can be laid out in array that is basically turned on its side to get the other thing, thus the commutability of multiplication.

 

However, emphasizing that one notation is right or wrong feels like it punishes young kids for understanding. It's like taking a beautiful, detailed story written by a 7 yo and tearing it apart for a couple of spelling and mechanics errors. Even more so if you've got a kid who spelled it the British way and you're tearing that apart.

[Guinevere]

Maybe if you think about it in terms of our spoken language, it would make more sense?  If I build something with two-by-fours, it's written 2x4's.  When reading math problems, when you see an x, you read "of".  So, either "by", or "of" could be replaced for the x.  2 of 5 is 2 sets of 5 objects.  1x2 is 1 of 2, or 1 set of 2 objects.  If you want to use "by", 2 by 5 is 2 sets, by 5's.  So 5, 10.  4x6 would be 4 by 6 or 6, 12, 18, 24.  

 

ETA That while I understand it is the same amount either way, and different resources may teach it either way, I prefer to keep in mind the "by" and "of" phrases associated with multiplication.  Then, my kids know when someone says they need rows of chairs set up, 8 by 5, they mean 8 rows, 5 chairs each.  Also, they know, nearly without thinking, that the "by" can be replaced with 8x5, and they need 40 chairs.  

Edited June 3, 2017 by Guinevere

[Hunter]

I don't fully understand it all. I'm only an outsider that has looked in at people who did understand this stuff, but when notation is different, what can be done with that notation is different.

 

http://io9.gizmodo.com/5977095/why-we-should-switch-to-a-base-12-counting-systemI'm not sure how much changes when the base is changed.

 

I do remember a bit about the Roman and Greek math my son worked on. The Roman numeral system did not allow many of the algorithms we are familiar with today. The Koine greek letters he was using didn't allow algebra. So other methods were used.

 

When translating spoken languages, it is not always as simple as a word for word translation.

 

Math is not any more fixed that phonics. We make a lot of choices. and the later choices are based on earlier choices. Not everything is as fixed as people think it is.

 

After all these years, the Koine Greek Euclid pages are still up. Not that I'm suggesting anyone teach this. :lol:

http://mysite.du.edu/~etuttle/classics/nugreek/contents.htm#conts

[fralala]

This thread should be called "Interesting Math Question." I don't really have much to add except regarding your quest for a visual/chart that demonstrates the times tables. Do you like the multiplication squares? Those are my favorite for the steps they make a person go through, and the patterns they demonstrate.

[maize]

I also see this as a matter of understanding language. One time, two times, three times--these words have meaning. I've been to Italy three times. "Three times" is how many occurences or iterations of something. In this case "trips to Italy" is the something. I could say "I have been three times on trips to Italy".

 

If I make a group of five three times I have three times five.

 

I don't see linguistic arbitrariness here. "Three times" tells you how many, "five" tells you of what--groupings of five (though trips to Italy are more fun).

 

Symbols (x or * or · or () ) represent language-driven concepts; the word times has actual meaning in the English language.

[winterbaby]

I agree about the meaning of "times," but people ignore this to the point that may no longer be descriptively true of their English usage. They treat "times" as a verb-like word meaning the first number expands the second number of times. Not helped by the teachers all saying "any number times one is itself."

[Noreen Claire]

Coincidence that this showed up in my email today? It's from the lead authors of the math Common Core standards... 

 

http://commoncoretools.me/2017/06/07/ways-of-thinking-and-ways-of-doing/