Mathy Folks: 4x7 or 7x4? Does it matter?

[vonbon]

Hello Fellow Boardies!

 

Can someone who loves math help me out with this?  DH7 and I are exploring multiplication.  She is understanding the basics and figuring problems out with a variety of methods (mental math, adding, using counters, counting by 1's, 2's, 5's, fact memorization, etc.)  I kind of let her arrive at the answers in her own way and then introduce other possible ways of solving things.

 

When I lay out counters in an array or write out a problem, does the order of the numbers matter?

 

For example, we have 4 boxes with 7 oranges in each box.  Should that be written 4x7 or 7x4?  

 

I know that for the answer / product, the order doesn't matter (Commutative Property.)  But I would like to be labeling things accurately if it will be important for her in higher math concepts in the future.  I am trying to lay a good / accurate foundation / number sense.

 

How would you label this array below in regards to rows and columns?  I would say this is 7 rows and 4 columns.  7 rows with 4 in each row.  So would it be incorrect to "label" it on paper or with words as 4x7?  

 

x x x x 

x x x x 

x x x x 

x x x x

x x x x 

x x x x 

x x x x 

 

Does it matter if you label it 7x4 or 4x7?  Is it just a matter of personal perspective?  Should I be matching up rows/columns in arrays with how the problem is written?

 

Also, when you begin to talk about columns and rows (like in populating tables via Excel data or the x and y axis) at some future point, will the order of the numbers matter?  Or would one just assign the labels of X and Y to the data and plug it in accordingly?

 

I might be mixing concepts--sorry.  Please weigh in with any helpful thoughts and please be kind.  I've never taught 2nd/3rd grade math before and I think I'm mixing in things from basic up through college math and science data.  My main goal is to lay a good foundation for future math but I'm not super-adept at higher math myself.  It's all a bit fuzzy.  Thanks.

[Lori D.]

Not a mathy person, but I'll just toss in that while the product (answer) is the same, the order does mean something a little different -- the first number is the "multiplier", the second number is the "multiplicand" (set):

 

4x7 means 4 sets of 7s -- so you are talking about an operation you are doing to a set of 7

7x4 means 7 sets of 4s -- so you are talking about an operation you are doing to a set of 4

 

(Note: in equally many places, 4x7 is read this the OTHER way, to mean "4 taken 7 times", or "4 sets multiplied by 7", so in this version, 7 is the multiplier and 4 is the multiplicand (set)). So reading it either way appears to be correct; just teach it consistently to what fits with your math program. It appears the way I initially labeled with multicand 1st and multiplier 2nd is more typically US; labeling with multiplier 1st and multiplicand 2nd is more typical to UK and Canada.) 

 

Again, as far as the product (answer) is concerned, due to the Commutative Property of Multiplication, the order of the factors does not matter -- both "groupings" (4 sets of 7 or 7 sets of 4) yield a total of 28. As far as your rows and columns… Because in English, texts are read horizontally left to right, I would say that in your chart of x's, 4 across is your "set" and you've stacked 7 sets up so 7 is your multiplier, so I'd say that was 7x4 (7 sets of 4). For 4x7 (4 sets of 7), you would have 4 rows, each with 7 x's. For demonstration purposes, I'd do both, and also circle sets both horizontally and vertically so you're showing how interconnected 4, 7, and 28 are.

 

A different approach might be to think of these 3 numbers together as a "fact family" -- there is an interconnectedness between the numbers 4, 7, and 28, and with multiplication and division. Cut a triangle out of an index card and put each of the three numbers on a corner:

 

………28

 

…..4….….7

 

Depending on what "order" you "read" the numbers in, yields a different math fact:

4x7= 28

7x4=28

28/4=7

28/7=4

 

 

 

Hope that helped, and didn't just add to the confusion. ;) Warmest regards, Lori D.

Edited April 29, 2016 by Lori D.

[TKDmom]

No, it doesn't really matter, because you can word things either way. If you have a matrix, it can be either 7 rows of 4 or 4 columns of 7.

When we used Math Mammoth, the text made a big deal about writing multiplication problems in the "right" order. I found it confusing and annoying. I think it's better to emphasize the commutative property and label manipulatives in whatever way makes sense to you.

[vonfirmath]

My son is in 3rd grade.  They would put a picture of an array like that out there and expect him to write ALL 4 of the multiplication and division associated problems. I forget what they call it "Number family"?

 

So the answer would be

4x7=28

7x4=28

28/7=4

28/4=7

 

[MrSmith]

No, it doesn't really matter, because you can word things either way. If you have a matrix, it can be either 7 rows of 4 or 4 columns of 7.

When we used Math Mammoth, the text made a big deal about writing multiplication problems in the "right" order. I found it confusing and annoying. I think it's better to emphasize the commutative property and label manipulatives in whatever way makes sense to you.

 

 

Actually, in the case of an actual matrix, order does in fact matter.  That is, a 4 by 7 matrix is not the same as a 7 by 4 matrix (in fact the first is the transpose of the second, and vice versa).

 

More to the point of the OP's question, when dealing with multiplication at this level, whether it's written 4x7 or 7x4 doesn't so much matter from a pragmatic perspective - the product is the same.  However, at a deeper conceptual level, it's nice when the student actually understands that "4 groups of 7" (4x7) is distinctly different from "7 groups of 4" (7x4).

 

Indeed, one can hear the difference in the naming of the thing: "4 times 7" = "7 + 7 + 7 + 7" vs "7 times 4" = "4 + 4 + 4 + 4 + 4 + 4 + 4"

 

(As an aside, this point becomes even more salient in computer science when dealing with array indexing.  Since all data memory is essentially flat, a 'matrix' has no real meaning, and is defined in terms of rows and offsets).

 

[ForeverFamily]

Not a mathy person either but just in case this helps...an easy way that I remember the difference between 4x7 and 7x4 is by replacing in my head the multiplication symbol with the word "of." So 4 of 7's, and 7 of 4's. I have taught my kids this by using the words "times" and "of" interchangeably, they have picked up on the difference fairly easily this way. But they also understand the commutative property from using manipulatives like c-rods. Not sure if any of that made any sense, lol. :)

 

Edited to add: A great resource that really helps me better understand how to teach math are the Education Unboxed videos. Best of all they are free.

 

http://www.educationunboxed.com

Edited April 30, 2016 by ForeverFamily

[scoutingmom]

And Rightstart does it the other way.... 4x7 is 4 taken 7 times, so 4 in a group and 7 groups.

 

Sent from my SM-T530NU using Tapatalk

[vonbon]

More to the point of the OP's question, when dealing with multiplication at this level, whether it's written 4x7 or 7x4 doesn't so much matter from a pragmatic perspective - the product is the same.  However, at a deeper conceptual level, it's nice when the student actually understands that "4 groups of 7" (4x7) is distinctly different from "7 groups of 4" (7x4).

 

This sums up my question exactly--thanks.  I think on "a deeper conceptual level" about many things in life.  Somehow straightening this concept out in my head will help me teach my daughter, even if she doesn't yet need to know it.  Thank you!

[vonbon]

Not a mathy person, but I'll just toss in that while the product (answer) is the same, the order does mean something a little different -- the first number is the "multiplier", the second number is the "multiplicand" (set):

 

Lovely!  Thanks, Lori D.  I was googling earlier today to figure out those labels: multiplier, mutiplicand.  I knew they existed--just couldn't remember them.  I'll go back and read your entire response more thoroughly.  Thank you!

[vonbon]

Technically, yes, it does matter.  And it comes into play in higher math levels. For a seven-year-old (am I reading that right?), it really doesn't matter, and actually, learning the fact that multiplication is commutative is a good thing to learn at that age.

 

OK, super!  Thank you!  Yes, she's 7, so my question is probably more to put my own mind at ease and make sense of it. This has been nagging me for about a month now.  Reading your response, I see that I can probably let this go for now unless I'm writing out or setting up an array with manipulatives.  Then, I might as well be accurate about it.  But for DD, I will focus on getting the commutative property across.  

Edited April 30, 2016 by vonbon

[kitten18]

Not a mathy person either but just in case this helps...an easy way that I remember the difference between 4x7 and 7x4 is by replacing in my head the multiplication symbol with the word "of." So 4 of 7's, and 7 of 4's. I have taught my kids this by using the words "times" and "of" interchangeably, they have picked up on the difference fairly easily this way. But they also understand the commutative property from using manipulatives like c-rods. Not sure if any of that made any sense, lol. :)

 

Edited to add: A great resource that really helps me better understand how to teach math are the Education Unboxed videos. Best of all they are free.

 

http://www.educationunboxed.com

Yes, I love that!  

We always say "of" when multiplying.

I think it helps a lot when you get to fractions. When they see 1/2 x 1/4 and can say in their head "one half of one fourth" then it makes sense that it's 1/8.

Of course this would work the other way too, one-quarter of one-half. So it probably didn't apply to the original question.  :tongue_smilie:

[vonbon]

(Note: in equally many places, 4x7 is read this the OTHER way, to mean "4 taken 7 times", or "4 sets multiplied by 7", so in this version, 7 is the multiplier and 4 is the multiplicand (set)). So reading it either way appears to be correct; just teach it consistently to what fits with your math program. It appears the way I initially labeled with multicand 1st and multiplier 2nd is more typically US; labeling with multiplier 1st and multiplicand 2nd is more typical to UK and Canada.) 

 

Great!  Thanks for this as well.  I didn't want to confuse people, but I almost put this as part of my question in the OP.  It seems like sometimes I'll come across a video or website or something out of the UK on math and will hear them say things in an opposite order than commonly said in the US.  So I guess you have to consider the source and the way it's phrased.  

 

We're using Singapore.  I'm sure the TM goes into this somewhere, but I really appreciate the replies, as it's saved me a lot of time trying to find a reputable source that addresses this exact issue for both primary and higher maths.  

[domestic_engineer]

(Note: in equally many places, 4x7 is read this the OTHER way, to mean "4 taken 7 times", or "4 sets multiplied by 7", so in this version, 7 is the multiplier and 4 is the multiplicand (set)). So reading it either way appears to be correct; just teach it consistently to what fits with your math program. It appears the way I initially labeled with multicand 1st and multiplier 2nd is more typically US; labeling with multiplier 1st and multiplicand 2nd is more typical to UK and Canada.) 

This is interesting information!  I didn't know this!  Thanks for sharing.

 

And Rightstart does it the other way.... 4x7 is 4 taken 7 times, so 4 in a group and 7 groups.

 

Sent from my SM-T530NU using Tapatalk

Yes!  I too wondered about why Rightstart chose to say 4x7 as 4 taken 7 times.  

Dr. Cotter responded with the following text that I found insightful and helpful:

 

Sometimes 6 × 3 is thought of as “6 groups of 3.†However, consistency with the other arithmetic operations requires a second look. When adding 6 + 3, we start with 6 and transform it by adding 3. When subtracting 6 – 3, we start with 6 and transform it by removing 3. When dividing 6 ÷ 3, we start with 6 and transform it by dividing it into 3 groups or into groups of 3s. Likewise, 6 × 3 means we start with 6 and transform it by duplicating it 3 times.

 

In the array (an arrangement of quantities in rows and columns) model, 6 × 3, 6 represents the horizontal quantity and 3 the vertical quantity. This is also consistent with the coordinate system; in (6, 3), the first number, 6, indicates the horizontal number and 3, the vertical number.

 

Unfortunately, as the poster pointed out, it does not agree with the naming of matrices in linear algebra. In this case, I think arithmetic should take precedence.

 

[SparklyUnicorn]

I think it's ok to point out the difference, but to emphasize that the answers are the same.  That's a basic property of multiplication.  No need to make this seem complicated and confusing though. 

 

 

[maize]

I teach my children that 3x7 means three groups of seven.

 

This matches up with the way we use "times" in English in other cases; I can run around the block three times, fill a two cup measuring cup three times, call my friend three times. Three times means three iterations of something, in this case, three iterations of seven.

Edited April 30, 2016 by maize

[Sarah0000]

My DS has also recently started multiplication and I've found it easier to teach that the first number is the number of groups. That's because I want him to really focus on the groups, not the amount in each. Since we are also doing division at the same time its helped to remind him to separate the number into groups. Groups, groups, groups. I'm finding it helps to separate his thought processes away from adding and subtracting to thinking in groups by always saying the number of groups first. When you look at a picture or real life arrangements you notice the number of groupings first. So, first number is groups. Later when this becomes more automatic it makes sense to explain it both ways, or either way.

 

Are you in Singapore 2a? IIRC, the illustrations went both ways, at least it seemed that way to me and how I naturally looked at the groupings.

 

BTW, I'm not an expert and this is my first child. Its just what I've found to be true for the very beginning of multiplication/division.

[wapiti]

There is an article discussing this:  https://medium.com/i-math/why-5-x-3-5-5-5-was-marked-wrong-b34607a5b74c#.k7mcp4ur8  There is an interesting discussion in the comments; I would disagree with the author (and agree with the comments that disagree with the author).

Edited April 30, 2016 by wapiti

[daijobu]

  Please weigh in with any helpful thoughts and please be kind.  I've never taught 2nd/3rd grade math before and I think I'm mixing in things from basic up through college math and science data.  My main goal is to lay a good foundation for future math but I'm not super-adept at higher math myself.  It's all a bit fuzzy.  Thanks.

 

I'm going to weigh in on the "please be kind" part.  You have asked an excellent question, and I want to encourage you to keep asking questions of this sort on these boards.  You are doing your students a great service by being completely 100% clear on what is going on with even these basic operations.  Good for you!   If you encounter another topic along these lines, please keep asking; the pp's and others will provide lots of great ideas.  

 

I'm going to add another bit that may or may not be helpful.  In addition to arranging your counters into a 4x7 array, you can take those same counters and arrange them into a 2x14 array.  Or a 1x28 array.  Because all those operations provide the same result.  

 

Have fun and go deep whenever possible!  

[ScoutTN]

Interesting question!

 

I am not mathy and have taught my kids as Right Start teaches, that 4x7 is 4 taken 7 times. The arrays are read across times down.

I have noticed that Math Mammoth teaches the opposite way.

 

Edited May 1, 2016 by ScoutTN

[scoutingmom]

I just remember that one year I had a teacher that was very specific that they were a specific order. I didn't see the point, but it was very important to that one teacher. I don't even remember which order it was.

 

Sent from my SM-T530NU using Tapatalk

[heartlikealion]

More importantly, who has 28 oranges? Did that many come in a bag? ;)

 

Looking at your picture, I would say 7 rows because rows go across to me (think crossword puzzle if nothing else) and columns down. But if I actually had 7 rows and 4 in each row sitting in front of me, I would describe it as 7x4. That's just me. I don't have anything to back this up, just this is how my brain works.

 

We'll be going into multiplication soon, just barely touched on it. We bought the Times Tales DVD download. So basically I'm saying I don't have a BTDT experience to share, just thinking that you may be overcomplicating it. Of course, I don't know what higher math is affected by swapping things. The highest math I took was trig.

[Tsuga]

On a phone so if this sounds curt my apologies.

 

Rows come first, then columns. So 4x7 is four rows of seven each. The multiplier is the number of rows, multiplicand is the number of columns.

 

So 4x1 is a column of four; 1x4 is a row of four.

 

Keeping this in mind helps in much higher math; I suspect knowing that it goes rows then columns would have helped me earlier as well.

 

The reason 3x4 and 4x3 can be different is because the numbers occupying the spaces in the rows and columns might be different. So if it is three cows by four cows, no biggie.

 

However if you have a matrix of cows and pigs, the arrangement could be different. Let's say you first ask your pigs and cows to line up (from a straight line) in three rows of five (3x5). They are in a line that goes pig pig cow cow cow pig cow pig cow cow cow cow pig pig pig.

 

The matrix looks like:

 

Pig pig cow cow cow

Pig cow pig cow cow

Cow cow pig pig pig

 

Is not the same as lining them up 5x3 from the same line:

 

Pig pig cow

Cow cow pig

Cow pig cow

Cow cow cow

Pig pig pig

 

Notice that even if you turn the second matrix 90 degrees to look the same as the first, the arrangement of cows and pigs is different.

 

This becomes important in statistics and computer programming.

Edited May 2, 2016 by Tsuga

[maize]

I'd be interested to know how this is conceptualized internarionally as well, I think math language is more ambiguous in English than in many other languages.

 

I know in France it is definitely multiplier (number of groups) first then multiplicand (number of items in each group), I believe this is true in Spanish speaking countries as well.

 

Does anyone know about other countries?

 

(Tsuga, do you have multiplier and multiplicand reversed in your post above?)

Edited May 2, 2016 by maize

[rutheart]

The order is important any time you're talking about dimensions of a physical object. A 3x6 door looks nothing like a 6x3 door.

 

Ruth

[maize]

The order is important any time you're talking about dimensions of a physical object. A 3x6 door looks nothing like a 6x3 door.

 

Ruth

Well now, ain't that true?!? Edited May 2, 2016 by maize

[Tsuga]

I'd be interested to know how this is conceptualized internarionally as well, I think math language is more ambiguous in English than in many other languages.

 

I know in France it is definitely multiplier (number of groups) first then multiplicand (number of items in each group), I believe this is true in Spanish speaking countries as well.

 

Does anyone know about other countries?

 

(Tsuga, do you have multiplier and multiplicand reversed in your post above?)

Ack yes! I will reverse. I knew I should not have answered on a phone. My brain shrinks to fit the screen.

 

Edit: it is fixed now.

Edited May 3, 2016 by Tsuga

[Loesje22000]

I'd be interested to know how this is conceptualized internarionally as well, I think math language is more ambiguous in English than in many other languages.

 

I know in France it is definitely multiplier (number of groups) first then multiplicand (number of items in each group), I believe this is true in Spanish speaking countries as well.

 

Does anyone know about other countries?

 

(Tsuga, do you have multiplier and multiplicand reversed in your post above?)

3x7 is 'drie maal zeven'

So three sevens.

[ScoutTN]

On a phone so if this sounds curt my apologies.

 

Rows come first, then columns. So 4x7 is four rows of seven each. The multiplier is the number of rows, multiplicand is the number of columns.

 

So 4x1 is a column of four; 1x4 is a row of four.

 

Keeping this in mind helps in much higher math; I suspect knowing that it goes rows then columns would have helped me earlier as well.

 

The reason 3x4 and 4x3 can be different is because the numbers occupying the spaces in the rows and columns might be different. So if it is three cows by four cows, no biggie.

 

However if you have a matrix of cows and pigs, the arrangement could be different. Let's say you first ask your pigs and cows to line up (from a straight line) in three rows of five (3x5). They are in a line that goes pig pig cow cow cow pig cow pig cow cow cow cow pig pig pig.

 

The matrix looks like:

 

Pig pig cow cow cow

Pig cow pig cow cow

Cow cow pig pig pig

 

Is not the same as lining them up 5x3 from the same line:

 

Pig pig cow

Cow cow pig

Cow pig cow

Cow cow cow

Pig pig pig

 

Notice that even if you turn the second matrix 90 degrees to look the same as the first, the arrangement of cows and pigs is different.

 

This becomes important in statistics and computer programming.

 

 

Ok, this makes sense to me.

SO, why do Right Start and some other programs teach it the other way if students will need to reverse their understanding when they get to higher math?

[ScoutTN]

Think of it this way:

 

I'm making my grocery list, and I write out "milk x3" which means I need three gallons of milk, or milk three times.

 

Likewise, 7x3 means I need 7 three times.

 

 

This makes sense to me too. This is how my mind thinks of it.

But this is the opposite of what Tsuga said. I understand her answer, I think.....

 

Are students who learn it as multiplicand x multiplier going to be confused for later math?

[domestic_engineer]

Ok, this makes sense to me.

SO, why do Right Start and some other programs teach it the other way if students will need to reverse their understanding when they get to higher math?

 Based upon Dr. Cotter's response (see my post above) ... it's better/easier for a 7-10 year old to stay consistent with the other operations of arithmetic that they are learning than to worry about matrices and linear algebra down the road.  Once you are to the point of dealing with linear algebra and matrices, you should be able to deal with the inconsistency and quickly adapt to the notation of linear algebra.   (haha - I'm imagining a college student exploding and throwing a tantrum like my tween!)

[vonbon]

Yes, the labels will matter. The dependent variable is graphed on the y axis and the independent variable on the x axis.

Y is a function of x..or y depends on x, symbolically y=f (x). The elementary teachers here do discuss that in words with those who are interested when they do their data unit, without going to symbols.

 

OK, I remembered something like this.  Probably from research work (science degree but haven't worked in that field in a long time.)  And now I'm remembering that this doesn't come up until middle school / high school math...so we have a while.  :)

[heartlikealion]

 Based upon Dr. Cotter's response (see my post above) ... it's better/easier for a 7-10 year old to stay consistent with the other operations of arithmetic that they are learning than to worry about matrices and linear algebra down the road.  Once you are to the point of dealing with linear algebra and matrices, you should be able to deal with the inconsistency and quickly adapt to the notation of linear algebra.   (haha - I'm imagining a college student exploding and throwing a tantrum like my tween!)

 

I can't think of a time in algebra where any of this came up. Just speaking of my own experience, that is.

 

ETA: of course this could be a memory thing or I was in la la land...

 

Edited May 2, 2016 by heartlikealion

[vonbon]

My DS has also recently started multiplication and I've found it easier to teach that the first number is the number of groups. That's because I want him to really focus on the groups, not the amount in each. Since we are also doing division at the same time its helped to remind him to separate the number into groups. Groups, groups, groups. I'm finding it helps to separate his thought processes away from adding and subtracting to thinking in groups by always saying the number of groups first. When you look at a picture or real life arrangements you notice the number of groupings first. So, first number is groups. Later when this becomes more automatic it makes sense to explain it both ways, or either way.

 

Are you in Singapore 2a? IIRC, the illustrations went both ways, at least it seemed that way to me and how I naturally looked at the groupings.

 

BTW, I'm not an expert and this is my first child. Its just what I've found to be true for the very beginning of multiplication/division.

 

I like what you and other posters are saying about groups and then what's contained in each group,  I get "Milk x 3"; very helpful example (above.)  This is helpful.  I'll try to phrase it that way: "We have 4 groups and there are 7 in each group."  I can see using this verbiage too: "Seven groups of four."  It's probably best if a student can hear it said or see it written all sorts of ways and get the concept.  

 

We're just a few units into Singapore 2B.  I like it a lot, but sometimes I go "off road" and use manipulatives, Family Math, or just fun, real-life examples or what I call "story math" (outrageous word problem stories with silly drawings and colorful characters made up on the fly) to get away from the pages upon pages in Singapore and to casually "test" a bit and see if DD truly understands what she's learning there.  I get a bit restless using just straight Singapore TM, text, and workbook.

 

No expert here and this is my first child as well.  :)

[vonbon]

I'm going to weigh in on the "please be kind" part.  You have asked an excellent question, and I want to encourage you to keep asking questions of this sort on these boards.  You are doing your students a great service by being completely 100% clear on what is going on with even these basic operations.  Good for you!   If you encounter another topic along these lines, please keep asking; the pp's and others will provide lots of great ideas.  

 

I'm going to add another bit that may or may not be helpful.  In addition to arranging your counters into a 4x7 array, you can take those same counters and arrange them into a 2x14 array.  Or a 1x28 array.  Because all those operations provide the same result.  

 

Have fun and go deep whenever possible!  

 

Thank you!  "Please be kind" is because I have a tendency to "go deep" in many areas of life and it can tend to drive people a bit nuts, especially those who are more pragmatic or who do not want to or don't have the time to slow down and analyze or ponder.  I prefer not to skim the surface with most things.  I understand that this tendency can be tiresome to some and sometimes I sense that it can be taken as being pretentious.  That's not my motivation--I don't have time for that!  ;)  

 

When I can clearly conceptualize things in my visual brain and grasp them through and through on a deep level, it gives me more confidence that I'm teaching correctly and laying a good foundation of number sense with my kiddos.

 

We are just getting into this: "In addition to arranging your counters into a 4x7 array, you can take those same counters and arrange them into a 2x14 array.  Or a 1x28 array.  Because all those operations provide the same result."

[vonbon]

More importantly, who has 28 oranges? Did that many come in a bag? ;)

 

Looking at your picture, I would say 7 rows because rows go across to me (think crossword puzzle if nothing else) and columns down. But if I actually had 7 rows and 4 in each row sitting in front of me, I would describe it as 7x4. That's just me. I don't have anything to back this up, just this is how my brain works.

 

We'll be going into multiplication soon, just barely touched on it. We bought the Times Tales DVD download. So basically I'm saying I don't have a BTDT experience to share, just thinking that you may be overcomplicating it. Of course, I don't know what higher math is affected by swapping things. The highest math I took was trig.

 

Our friend has orange trees and brought us a couple of huge bags of them awhile back.  I kept one in the garage because it had about 50 oranges in it and it's cooler in there.    :laugh:   We're finally through the second bag!

 

So yesterday at the breakfast table we did...orange math!  We pretended to pick them, haul them to the table in a sack, and then we divided them up between ourselves.  It worked well to truly let Division sink in to our brains and then we moved on the the concept of "Division with a Remainder" with the oranges, leaving an extra one as a remainder.  My kids quickly said, "Yeah, but you can just cut it in half and split the remainder."  So...that leads to another upcoming concept in Singapore: Fractions.    

 

Math with Vitamin C!

[vonbon]

On a phone so if this sounds curt my apologies.

 

Rows come first, then columns. So 4x7 is four rows of seven each. The multiplier is the number of rows, multiplicand is the number of columns.

 

So 4x1 is a column of four; 1x4 is a row of four.

 

Keeping this in mind helps in much higher math; I suspect knowing that it goes rows then columns would have helped me earlier as well.

 

The reason 3x4 and 4x3 can be different is because the numbers occupying the spaces in the rows and columns might be different. So if it is three cows by four cows, no biggie.

 

However if you have a matrix of cows and pigs, the arrangement could be different. Let's say you first ask your pigs and cows to line up (from a straight line) in three rows of five (3x5). They are in a line that goes pig pig cow cow cow pig cow pig cow cow cow cow pig pig pig.

 

The matrix looks like:

 

Pig pig cow cow cow

Pig cow pig cow cow

Cow cow pig pig pig

 

Is not the same as lining them up 5x3 from the same line:

 

Pig pig cow

Cow cow pig

Cow pig cow

Cow cow cow

Pig pig pig

 

Notice that even if you turn the second matrix 90 degrees to look the same as the first, the arrangement of cows and pigs is different.

 

This becomes important in statistics and computer programming.

 

I think I love this...It could be very helpful...

 

But I'll have to look at the multiplier / multiplicand and adjust this?   Hmm....

 

 

Edited May 2, 2016 by vonbon

[Tsuga]

Ok, this makes sense to me.

SO, why do Right Start and some other programs teach it the other way if students will need to reverse their understanding when they get to higher math?

Mystery of the universe. I do not know. I got that education as well. :(

[Tsuga]

I think I love this...It could be very helpful...

 

But I'll have to look at the multiplier / multiplicand and adjust this? Hmm....

I fixed it based on subsequent comment.

[vaquitita]

This is something I've wondered too. I had noticed that Miquon teaches it 2 of 4. Multiplier X multiplicand. And then some traditional maths I looked at (study time, R&S) taught multiplicand X multiplier. Now I just finished MUS gamma with one of my kids and I have no idea which way they taught it. Lol. Now I will have to go back and check. It did focus on how the answer is the same either way. Because if stuff e had done before, this was intuitive to my son and we didn't discuss which was which.

 

Eta: now that I think back I think MUS taught it multiplier X multiplicand. Because with the problem written vertically the bottom number was the'over' or how many in the row (multiplicand) and the top number was the 'up' of number of rows (multiplier).

Edited May 3, 2016 by vaquitita

[Ausmumof3]

Thank you! "Please be kind" is because I have a tendency to "go deep" in many areas of life and it can tend to drive people a bit nuts, especially those who are more pragmatic or who do not want to or don't have the time to slow down and analyze or ponder. I prefer not to skim the surface with most things. I understand that this tendency can be tiresome to some and sometimes I sense that it can be taken as being pretentious. That's not my motivation--I don't have time for that! ;)

 

When I can clearly conceptualize things in my visual brain and grasp them through and through on a deep level, it gives me more confidence that I'm teaching correctly and laying a good foundation of number sense with my kiddos.

 

We are just getting into this: "In addition to arranging your counters into a 4x7 array, you can take those same counters and arrange them into a 2x14 array. Or a 1x28 array. Because all those operations provide the same result."

Fwiw homeschooling has really deepened my understanding in so much stuff. I was always good at following the rules as taught but I now understand so much more of the reasons behind the rules.

[Happygrl]

Threads like this are one of the many reasons I love this board.  Most HS boards/groups I've been part of would have looked at me like I had two heads for asking a question like this and say it obviously doesn't matter.  

 

I never thought to wonder about this...but I'm glad I know now!  Thanks for asking it, OP.

[vonbon]

Threads like this are one of the many reasons I love this board.  Most HS boards/groups I've been part of would have looked at me like I had two heads for asking a question like this and say it obviously doesn't matter.  

 

I never thought to wonder about this...but I'm glad I know now!  Thanks for asking it, OP.

 

Maybe this is one reason I'm not connecting with a ton of parents / homeschoolers IRL.   :laugh:  Two heads!