Error with our number formats

Lesson on something that we all do but could be improved upon.

Some of this sounds weird but stick with me.

       So back in 2000 there was an organization called OECD (Organization for Economic Co-operation and Development) that started keeping track of student progresses in various educational formats (math science reading etc.)

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      The OECD is basically a forum where different governments around the world can share experiences and can work together to identify and hopefully solve common problems.  Think of the OECD like that person that comes into businesses to identify and help solve problems by using the experiences of other businesses.

      My personal analogy for this would be if the world was a school.  The United Nations is the group that works together to create global rules and does the mediating.  Interpol is the ones who enforce said rules.  Leaving the OECD who is kind of the secretary who keeps track of how everyone in the school is doing and offers help via statistics.

-You might think of the OECD differently then I do but moving on-

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~So there is a organization spying on my children's test scores! I took them out of standardize testing!

       Nope not the same thing  OECD keeps track of things like how many kids are graduating schools,  how many hours are spent on each subject,  how students are doing in the subjects all year round etc.   Here's an example; 

This is an example of OECD's
Programme for International Student Assessment. (or PISA)

~I knew Japan was higher but it can't be that higher!? can it?

Actually yes from 2009-2012 Japan was this much higher in overall scores.  This is however a math blog so I will only be going over why it is most likely that Japan is much higher then the US at math.  It definitely NOT what you are thinking.

~You don't know what I am thinking.....

Okay so what do you think is the reason then?

~Because in Japan, kids spend much more time in class learning about math.

       Actually no.  That isn't it at all.

I know I know more graphs sorry......

But in this graph it shows how many hours US, Germany, and Japan the average 8th grader spends in math / science classes per year.

So its not the teachers not getting enough time to teach kids math.

~So this is a pro common core vs old math comment then?

       Nope. Isn't that either.  It actually has nothing to do with schools or teachers this time.  It has to do with us as parents (and in some cases Preschool providers)  But before you take offend, it's not that you did something "wrong" it is that Japan figured out a much better way.

        So lets go back.... very basic math..... before addition....  what is the first form of math that we teach our kids.  I will give you a hint Elmo loves to sing about his ABC's and _____.

~Wait! How are we teaching our kids to count 123 wrong?

       It isn't about that we taught them wrong.  We were all taught this way.  Not the numbers in terms of 1, 2, 3, 4, 5, 6, 7, 8........  It's more of how think and say these numbers.

~What.....?

       It will all make sense in a moment.  But at least for a while we can stop blaming teachers, parents, schools, governments, etc about this for right now.  We will just blame it on the English language.

        No graph this time hurray!  But go look at these numbers and read them. Both in your head and out loud.  one two three four five six...........  It's safe to say that we all count them the same way because that is how we teach English speaking kids to count numbers.

       What if I was to tell you there was something weird in the way that we (and a bunch of other languages) do when counting that is kind of weird?

~I would say that you were dumb because it sounds fine to me

       Okay so in English we read/write everything left to right. Including numbers.  EXCEPT.... there is one time where we don't..... 11--->19

~How so?

       I'll explain.....  When we see "11"  we read and think of it as ELEVEN. right?  But if we stop to think about this where does that word come from?  when we read numbers shouldn't the numbers be in the word?  like "twenty-one or Fourty-eight".  Where did "eleven" come from?  Well folks we have an amazing thing in our life now called the INTERNET. So I looked it up.

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The word “eleven” is derived from the Old English word “endleofan” (pronounced “end-lih-fen”) which itself comes from the Germanic “ainlif,” a compound word: “ain” means “one” and “lif” was a version of the word “left.”   <-----   I googled

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       So basically we are teaching our kids that BOOM LEFT FIELD! a word that is still basically more German than English.  (granted there is nothing wrong with German)

~Wait.  Does that mean eleven roughly would translate as One left of One?

       Basically...  Which also means as of 2016 we do not actually have an English word for "11"  Just a mispronounced German word that was transferred through old English.

~Okay so 11 and 12 are technically not in English but how is any of this stuff not being read left to right?

       Well after 11 and 12.  We have 13 (thirteen), 14 (fourteen), 15 (fifteen), etc......  But lets think about how we are actually reading them.  All these numbers have a "1" first in line right?  but why do we read it as "FOUR teen" we would technically be reading it as

"4 after 10" Which if you think about this it makes our numbers in this area seem pretty weird.

~Fine! I read the teens places weird. How is that any different in Japan?

       Well my friend.  I have read during my various common core math research about how in Japan they read these numbers differently then us.  However, before reading this I double checked this with my friend Noel.  She has spent time during student exchange programs in Japan as well as she is VERY fluent in Japanese and English.  

*Thanks for the help Noel you are awesome*

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       The way you would say eleven in Japanese would be, "jyu-ichi"... literally translated into "ten one".  (jyu = ten) and (ichi = one)

~So your saying that we should teach our kids to say things like ten-one, ten-two, ten-three?  What next should I stop correcting my preschooler when they say "onety-one"?

       That is exactly what I am saying.  Granted that the idea of saying "ONEty-one" sounds like your just making up numbers BUT how is that any different then "FOURty-two".  The first one sounds made up but the second one is perfectly acceptable even though they are the same format of reading the numbers.  Long story short here is that we have been inconsistent with the format of how we read numbers.  Leading to a rocky foundation to build our math skills on.

This of course continues the same way into larger numbers. 

"ni-jyu-ichi"--->"two ten one" (21)... 

"ni-hyaku-ni-jyu-ichi" is "two hundred two ten one" (221).

etc....etc.....etc....... 

So think about the way these numbers sound.  "two ten one" vs "twenty one"....

~I don't hear that much difference.  It does kind of make sense here.

       Now if instead of reading the "2" in 21 as a "two" vs "twen" 

(I am too lazy to bother looking up where "twen" came from)

"twen-ty"  Or as it more closely sounds as "twen-T"

(Why can't we just make it easier on ourselves and keep it as the number it truly is. )

"two-ty"  Or as this more closely sounds as "two-T"

So the T sound that is prominent is kind of like a shortened mark stating that its in the TENS space.......get it......T........TEN.

We already do it for all other areas in the tens place.

(twenT)(thirT)(fourT)(fifT)(sixT)(sevenT)(eightT)(nineT)

~5 of these wouldn't even change then?

Exactly!  Only the 3 out of this list would be effected by changing its weird nickname

(twenT = twoT) (thirT = threeT) (fifT = fiveT)

~So this change would benefit us all how? And what about larger numbers? do they have to change?

      By creating a level of consistency.  All our numbers would then follow the exact same format.

       And no. All our other numbers are already in the same format as Japan.

 examples; 

3672 (read it out loud and in your head without using the word "and")

 Three thousand six hundred seventy two

(Three thousand) (six hundred) (seven T) (2) <----English

(Three thousand) (six hundred) (seven ten) (2)<---Japanese

Basically the same here.

3619 (read it out loud and in your head without using the word "and")

  Three thousand six hundred nineteen

(three thousand) (six hundred) (nine-teen)<--------English (but we read as "3-6-9-1")

(three thousand) (six hundred) (ten) (one)<---------Japanese           (read as "3-6-1-9")

~OH SNAP.  I taught my kid to count while trying to keep track of format changes.

       Don't worry I also did this same thing to my son and had no idea!

       Basically what I am trying to say here is when we start having kids do math problems with numbers.  Their little brains see the numbers as 19x6.  But what they are reading it as would look like this --> (9-1)x6.  As kids develop it gets easier to translate the numbers between how their written vs read. So how we are reading them as adults we don't even see it as a problem.

       I have been reteaching my son to count as onety one, onety two, etc.  He is 6, found it fun to count it a different way but was concerned that he would get in trouble at school for counting it that way.  I simply told him that this way creates a better foundation for learning and that I would handle any confusions with his teachers.  So if you agree that this new format makes sense and you teach your children this way.  Feel free to save this blog entry and show it to any confused: other parents, grandparents, teachers, principles, confused lady on the bus, etc........

       Because if you were to hear cashier say "that will come to two dollars and onety one cents" or "two dollars and one ten one cents" before you read this blog.  You could probably have understood what they were saying. (probably would have thought they were messing with you at first but you would have figured it out.) <---- This theory was tested.

Common Core Math Multiplication

Hello everyone reading this please keep in mind all my previous entries to my blog as they get kind of flipped with the most recent being on top..... SOOOOO like when you learn actual math you typically start with addition and subtraction first so feel free to start at the beginning.  But I will re explain some things as I come across things that are relevant.
And keep in mind that I do talk to myself as if I am someone else.  The does include arguing with myself.

Common core math misconception!

AAAAnyway I will start right into it with the picture to the right.
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Here we have an example of very basic understanding of multiplication.  What this teacher is attempting to teach is that the first number in the equation is the number of groups... With the second number being the amount in each group.

~But whats the difference the old way was better!

Actually my dear friend the teacher is teaching and grading this test based on the "Old way"  The old way is the numbers stayed where they are and DO NOT under any reason move.

~But there isn't a difference from 5 x 3 and 3 x 5.  You always get the same answer.

Hey look at you your already doing basic common core math!  Common core math points out there are different ways to look at and solve the numbers.  If you have read any of my previous blogs common core math teaches kids that there are TONS of ways to solve math problems and whatever way works best for you is still correct.

~Even the old way?

Yes! Even the old way....
This teacher grading this test was mistaken.  The students taking this test could have put them which ever way they wanted (5 groups of 3) or (3 groups of 5) etc.....
Because on question 2 when they drew out the rows it points out no matter which way you look at it is still has the same number of marks. (or the paper could have been turned to the side... the amount wouldn't change)

So the point I am making..... This is a misunderstanding issue by the teacher. NOT a common core math issue....

~Oh.....okay.

So lets think back to our childhood when we learned multiplication in schools.  Think about that big chart of numbers.  ~multiplication tables?  Exactly.

This is the basic idea for a multiplication table.... sure we can add colors and pretty pictures on it and whatnot.  BUUUUUT it still works based off one thing..... Memorization.  They really expected us at one time to memorize this thing!  I was given one of these charts at the end of 2nd grade.  That means I was 6 or 7 and expected to memorize this ridiculous chart.

~But I learned I could piece together this chart by counting by each number group in my head.  That's a common core skill of whatever way works best for you!  ~DARN IT! 

It was taught as "see this chart in your head and you will be able to solve all these questions just by looking at them and seeing the answer in your head."   Which a few kids could do. Obviously due to their memory capacity.  However the rest of us figured out tricks.  Or had good teachers that understood we couldn't visualize an ugly chart in our minds.

 So we counted by the numbers, Or had silly rhymes, or figured out silly tricks.
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examples;
6 and 6. picked up sticks and ended up at 36.
6 and 8 went through the gate and ended up at 48.
etc.etc.etc......

Or the 9's trick.

If you hold your hands out like this picture while your multiplying by 9's you quickly see 9's. (I am pretty good at mental math and its still faster just to use my hands like this)
Moving on...... if your question was 9 x 1. You then put your 1 finger down.  That gives you 9 fingers to the right of the finger you put down.  Answer = 9 ~Obviously!
But if your question was 9 x 7.  You put the 7 finger down.  That gives you 6 fingers on the left and 3 fingers on the right....... answer being........ 63 BOOM! instant solutions for kids.  Feel free to try this awesome trick out with all the other numbers between 1-10.
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~But the new way says we can't do those things!....

That's incorrect..... The common core way was made to give kids options.  Years ago in elementary school I got a bad math grade because I had a teacher who said I was "counting on my fingers still"  All because I figured out this trick with the 9's and I went from the slowest math student to the quickest at 9's.

The new way is "If this way is to confusing I can teach you different ways until we find one way that works best for you" 

~But teachers are failing kids for doing it a different way!

That's because they weren't taught the COMMON CORE idea of Common core. --------------------------------------------------------->


The main idea of common core is like giving directions.  You might prefer taking the highway. Its easier to understand but might take a bit longer.  While another person might be able to drive through the streets which is more complicated but gets them there faster.  If your a highway driver and your friend gives you 100 directions of going a ton of little side streets it makes sense that you might get lost.
But if you give your friend Highway instructions that other person might be like "BUT I COULD HAVE JUST GONE THIS WAY AND BEEN HERE SOONER!?"
(Both sets of directions get you to the same location and there might be tons more ways to get there that others might have preferred)

AAANYWAYS... back to math.

OLD METHOD                                                                      NEW METHOD

 ~But this makes my way seem much more complicated then it is!

Not really.  This is just the drawn out form of how you solve the problem with longform (old method). 
You multiply 3 x 7 as written on the right of the question. which gives you 21.  Obviously the old way says you can't have a 21 in the 1's place.  So instead you will pretend that the 20 is only a 2. (in the 10's place) and you write it above the place it belongs to as a place holder. (aka carrying the 2) 

The next step for the old way is you now need to multiply diagonally which is a weird concept in itself.  So 2 x 7 as drawn in the left of the equation making 14.....

But wait don't forget our tiny place holder.!  AND that place holder has to be added and not multiplied. 

Now most kids aren't always able to handle a double math question in their head so they would write it down.  As shown as 14+2. 
 Granted us adults are skilled enough to be like OH that's just (2x7)+2.  But we are teaching kids kids here remember.
Suddenly the "old" way doesn't seem like a short cut.  We just expect it to be shorter because we expect all kids to mental math all the easy parts.

-MOVING ON-

Box method......(as shown above)
Common Core math teaches that numbers can be broken apart.
(much like the old ways form of carrying numbers back and forth only without pretending its not what it is)
So 23 can be separated into 20.....and.......3 (because 20+3 = 23)
Its like pretending your carrying the 2 into the 10's spot. Only its a 20 because that's ACTUALLY what it is.
Then you do the two steps of multiplication just like when you did it before
7 x 3 (or 3 x 7) =21   and 7 x 20 (or 20 x 7) = 140
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~Whoa now STOP right there.  Kids this age can't multiply 7 x 20 that number is to high for them.

Fair enough but if you look back at my previous blogs you can split the numbers up even more;
 so two sets of 10's.  (7 x 10 = 70) and (7 x 10 = 70)  (70+70=140 same way as 7+7=14)

-or-

2 groups of 10's (10 x 2) x 7 <--- which can be replaced in any order and still come out the same.
7 x 2 = 14.......  Then 10 x 14 = 140
( or broken down even further. 10 x 10 = 100 and 10 x 4 =40.... 100 + 40 = 140)
SO many ways to break these numbers down.

~but this is soooooo much extra work.

Only until they can do it in their heads.  It teaches that numbers are fluid. Some brains see the streets.  While others see the highway.  Some kids find it easier to hold all these little numbers in their heads rather then be intimidated by the big numbers.
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BACKTRACKING!!!!!! 7 x 20.
Kids can count by 2's right? ~yea....  And kids can count by 10's right?  ~yea.....
and kids can count by 20's because it just combines those easy skills.  Instead of 2, 4, 6, 8, 10..... it is 20, 40, 60, 80, 100. Note the patter is the same but with the "0" for the 10's place.

-or you can combine the "old way" with the "new way" (yes its legal)
20   <-----its just 2 x 7 which is 14....but you add 0 to the end... making 140.
x7
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This seems like A LOT of extra work but keep track of whats easier for a kid to learn to do in their head.  Its not easy to keep track of carrying a 2 when your trying to do it in your head.
So common core math is teaching kids to learn understand math in their heads vs us as adults having to stop everything and find a piece of paper to figure everything out.
All this stuff is just extra steps that kids CAN take to better understand but do not HAVE to take.

~Okay back to the main question I get how kids can figure out 7 x 3 and 7 x 20.
So they are left with 140 and 21.

That's true.  Which we know gets added so 140+21 = 161.

~So if kids couldn't do 140+21 in their heads they could split it up into 100+40+20+1
which would be like adding 100+60+1 which is 161...... This is kind of easy.

But you don't have to do it that way if the "old way" is easier for you.
Just give it a fair chance to understand the idea of how numbers are fluid.

~Okay so I understand how this helps me on a math question but how does this relate to the real world... how would this help me day to day?

Well if you had a hallway you wanted to get carpets for..... and the dimensions was 7ft x 23ft. that looks too difficult to do in your head.
So instead break it up....only do the first 20ft of the hallway first. which is 7ft x 20ft.  Which would then be 140 square ft.  with the last chunk of the rug being 7ft x 3ft which would be 21 square ft.  Add those two rugs together to make 161 sq ft.

~But we already solved that one so it was easy to solve again.....

Okay how about this example then;

Old way                                                                                                         New way

The room you need a rug for is 23ft x 37ft

~but I would just tell the company those dimensions

But what about pricing..... you can see the price "by square ft" 

Basically you can split this room up into easy so solve chunks with mental math vs sitting and writing out the question. 

You can break up all the hard to count by little numbers away from the easy to count by bigger numbers.

(if you were to draw all the lines it would show the actual square ft and you could actually count them if you so chose to do it the long way)

20 x 30 = 600 (because 2x3 is 6 but you were counting by 20's{or 30's})

20 x 7 = 140 (like before)

30 x 3 = 90 (kind of like how 3x3 is 9 but one of the 3's was actually a 30)

7 x 3 = 21 (like before)

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So then you would just add the sq ft of the 4 rugs.

600 + 140 + 90  + 21    ~OH! and we could shift numbers around to best fit our brains!

Yep so it could be like this

600 +  100 + 90 + 40 +10 + 10 + 1 <--- combine the 10's with the 90 and the 40

600 + 100 + (90+10)+(40+10) + 1 <--- Now we made it easy to separate 100's 10's and 1's

(600+100+100) + 50 + 1

800 + 50 + 1 = 851

*If your having trouble understanding the addition break down read my blog entries about common core addition.  It will explain it better*

(This was a very difficult example)

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Here is a simpler version to see;

This wasn't explained very well in the picture but I will help.

Remember when I split 20 into (10 x 2) earlier...... SAME idea here.

20 x 4 is the question but the teacher wants the student to show a specific process which is to go through a point where there is 10 x 8.  If someone gave you that question with no explanation you'd be like WHAT IN THE WORLD!? but let me explain it.

-Some kids can do an easy question like 20 x 4 in their heads. (note about 20x7 earlier)

-Not all of them can and they break the numbers down and process them differently.

So the 20 as I showed earlier can be broken down into a (10 x 2) Thus the question would then be;

(10x2)x4 <----but its all the multiplication so the order doesn't matter. (similar to addition)

So you can solve (4x2)first... then x10.

that's where the "(2x4)x10" comes from (even though its not explained in this picture)

 

~Hey we just found the 8x10!

 

Yes we did! so 8 x 10 = 80! <--- because some people find 10's easier then 20's

 

 

 

NOW ON TO LATTICE METHOD MULTIPLICATION 

 -Disclaimer: I DO NOT like the lattice method personally.  I find it to look sloppy and you cannot do it in your head unless you are a super genius. But my sister likes it and so do a bunch of other people so I will explain it properly.......

 

 Okay to start explaining these are written out by steps to help me explain.

The question is 28x36 (or 36x28 doesn't matter) so 2- 2digit numbers which if put in this method would create a 2x2 square (note 2-8 on top and the 3-6 on the right)

Followed by the diagonal lines. (the order of which numbers you do first doesn't matter as long as you put the numbers in the appropriate places)

so the top left box explains setting up the problem.

The 2nd one next to it is starting the problem at the 8 and the 6 aka the 1's place (same as long form) 6 x 8 =48 so the 4 is in the 10's place and the 8 is in the 1's  they get separated by the diagonal line we made. (seems weird I know but it'll make sense in a moment)

You follow the same procedure with other 3 boxes. (2x6) then (8x3) then (2x3).

Now you would have the first box in the second line. 

(it was like your own mixed up multiplication table)

Now you follow the diagonal rows and add up the numbers. starting at the bottom right.

8 is in a group by itself.

4+4+2 is in a row which makes 10 so you would carry the 1 to the next diagonal row.

1+2+6+1 = 10 So carry the 1 again to the next diagonal row.

1+0 = 1.

then your answer is already written. answer = 1008

More digits in your numbers means a bigger square is needed so it tends to get more complex and confusing with larger numbers. So I do not like this method as I already said.

~Then why would you teach it?

Because I wouldn't fail someone for doing it this way. If they need to use a GIANT piece of paper to do a question like 1372 x 469223 then be my guest. However you need to do it.

Now for a more algebraic way to do this. (mainly for teaching adults or really smart kids who like playing with numbers)

28x36= (yes again)

You can break both numbers down if you wanted (yes both)

(20+8)x(30+6)= <-- its not all multiplication so you can just multiply them all this time like before......

you have to multiply the first part of the first number by both parts in the second number AND the second number in the first part by both parts in the second number. 

(this is another reason I explained the Lattice method because this is similar but without the ugly square drawings)

so first part of the first number 20

(20x30)+(20x6) = (600)+(120)

and the second part of the first number 8

(8x30)+(8x6) = (240)+(48)

Then adding all those numbers together

600+120+240+48 = 1008

This way is like doing the lattice method but without the box and without pretending that the 2 from the 28 isn't actually a 20.

~That seems SO DIFFICULT!

To you it might seem difficult but to others it might be the "easier way" but the end result is always the same.

So... Whatever way you feel like answering these math questions it fits into "common core math"  The teacher shouldn't fail you for doing the math wrong. They can only mark you down for not understanding that particular way.

As long as you give the method I am explaining a fair shot that's all I can hope to ask for.

The end result of common core math should be to have you excited to understand math in a way that clicks in your brain.  Instead of being depressed that the way you are being told to do it is like walking up hill while carrying bricks.

Anyway...... HURRAY MATH!!!