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!!!

Common core Subtraction continued

      This is a fun example of common core.  (nope I am not going to say it isn't common core)  This is in fact a perfect example of common core, it looks very intimidating in this explanation of it though so I will explain all the fun to you guys in a moment.

      So many (or at least my teachers taught my classes) of us were taught the "old way" of doing simple subtraction (10-9=) (7-4=) (etc....) two different ways.  The "normal" way of starting at the first number and counting down the number of places from the second number.
examples;
(10-9=) start at 10. Go down 9 places. 10,(9,8,7,6,5,4,3,2,1) (leaving you at 1 being the answer)
(7-4=) start at 7. Go down 4 places. 7,(6,5,4,3) (leaving you at 3 being your answer)

Following so far? awesome.

     The other way my teachers called the count up method.  It's where you would start at the second number and count up the number of places until you made it to the first number.
examples;
(10-9=) start at 9.     9, (10). you only moved up 1 space. Answer = 1 (same as before)
(7-4=) start at 4.     4, (5,6,7). moved up 3 spaces.  Answer = 3 (same as before)

     With this smaller scale of math the count up method of math is super easy.  Sure there would be math problems like (10-1=) which would be easier to start at 10 and count down.

 So with a bigger math problems; (examples;)
54-3=  (easier to count down from 54, 3 times) 54,(53,52,51) Answer = 51.
54-52= (easier to count up from 52 to get to 54) 52,(53,54) Answer = 2
*Both of these methods are still perfectly correct in common core math.*
     Also, the other way of writing out the problems is still correct. Examples;
   54       54
    -3     -52            (This way your just separating by 1s,10s etc.)
___    ____
   51        02

      Or if your a visual math person you can take out $0.54 and do these same problems by removing the change that your subtracting.
Lots of different ways already to do / teach/ learn math.  These are all the old ways BUT are all still considered "common core math"  Whichever way helps you to visualize and do math in your head.

"But Jared! what about when we have to borrow from other places in subtraction problems?"

     Happy you asked!  The only time the borrowing processes is easy to see is when its written on paper with one number on top of the other number. Otherwise it becomes very difficult to keep track of all those tiny edits to your original number.  This is because doing subtraction this way is a very linear process to math.  All the numbers are in blocks that you would then have to take from to move around (borrow from) and realistically create an equally annoying number to work with mentally.
EXAMPLE;
                                                                                   TENS  ONES
   57                                                                          5            7
  -38      (your basically seeing it like this)-->             -   3            8
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Yes, that is how I that problem when I write it out that way.
     Okay great!  So you start doing the problem but right away you notice you need to borrow.   You can't take 8 out of 7.  It would be like your friend taking away 8 of your 7 pennies when you still have dimes in your other pocket.  So you would borrow from the other pocket (in this case trading another friend 10 pennies for your dime)

     The problem now is you just dumped all of your pennies into one pile even though you only needed 1 penny from the pile you already had. Giving you 17 pennies. So now you have an entirely different double digit math problem in your math problem.

 5       7                       (then would become)    (5)4         17
-3       8                                                           -3           8
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"Yea Jared whats your point?"
   
     My point is that you are now stuck with 17-8.   A 2 digit number subtracted by a 1 digit number.  When the point of this processes was to separate 10's from the 1's places.
     So now your math problem inside your other math problem.  You now have to fall back on the other methods of subtraction. The only way said you either counted it up, or counted it down.  HOWEVER! if you have read my previous blog on subtraction you know there are other ways to do this problem.
The examples of other ways to do it would be to separate some of the numbers.

"BUT THATS SO MUCH MORE WORK!!!!!! "

     On paper yes it does look like more work.  BUT mentally (with a little practice) it becomes MUCH less work.

Examples; Parenthesis to show the break down of one number.
[10(+)7] would be more closely understood as [10 AND 7] <---because the combination would be 17.

17-8=                                                     17-8=
[10(+)7]-8=                                          17-[7(+)1]=
move 1 from 10 -->7                               Subtract the 7 first.
[9(+)8]-8=                                            17-7=10   Then subtract the remaining 1 from the answer
9+8-8=9                                                10-1=9

     On paper it seems like a LOT more work. However if you think about it in your head it is MUCH easier to do. (17-7-1=) VS (17... 16,15,14,13,12,11,10,9)

But I can do that problem in my head just by looking at it!

     This small of a question would be something to teach to a 1st or 2nd grader.  When you were 6-7 years old could you look at 17-8 and instantly know that it was 9?  (a few of you yes. the majority probably not and we would have to utilize some other way other then just memory)

     MOVING ON!...... now you know that    (5)4       17
                                                                              -   3         8             (17...16,15,14,13,12,11,10,9)
                                                                            __________
                                                                                   1          9

     Suddenly this way seems like it would be a lot more work to a kid trying to learn this way.  So maybe the written out version of "common core" math isn't as much "extra" work.

     Now for (one of many) common core ways of doing this problem

57-38= (same as before but this time we break down numbers)
[50(+)7]-[30(+)7(+)1]=
now the ones and tens are separated in a way that can be subtracted without borrowing.
50-30=20
7-7=0
1=1
turning your annoying math problem into;
20-1=19
(simplified version on paper would look like this)
57-38=
50-30=20
7-7=0
1=1
 20-1=19
(or as you develop your mental math skills you downsize the work into)
57-38=
20-1=19
         Because mentally removing the 7-7 from the equation is very easy. and 50-30 is easy also. Leaving only that remaining 1 at the end to subtract from 20.  BOOM! common core mental math short cut.

"That still never explained the math problem in the picture at the top of this post."

     Oh right! back to that.  54-27......  In that picture the "old way" looks like much less work then the new way.  BUT as I just finished pointing out, that is because all the shortcuts are added to the "old way"  but EVERY thing is written out in the "new way".

OLD WAY                                                                                   NEW WAY 
  5       4                                                                                  54-27=
-2        7                                       from math problem-->27+(3)=30 <-- nice even 10's
______                                                                               30+(20)=50 <-- closest 10's to 54
                                                                                               50+(4)=54 <--from math problem
(borrowing)                                                                        (20)+(3)+(4)= 27
                                                                              This can be done very easy in your head
  (5)4    14                                                                       (for vast majority of people)
-      2      7
________                                                                                      -OR-
                                                                            
14...13,12,11,10,9,8,7                                                            54-27=
                                                                                            27+(3)= 30+(20)= 50+(4)=54
   (5)4    14                                                                               20+3+4=27
-       2      7
________
         2       7
Answer:27

Too many different numbers to
keep track of with borrowing
process.
(only very few people can do mentally)


     So basically without shortcuts of mental math here and there it all looks like TONS of work.  The more mental math the less work it seems like.  "Common Core" math promotes the process of mental math therefore in the near future of using it you will pretty much drop most if not ALL of the actual work behind it and just be able to do it mentally.

 The old way is not wrong........I REPEAT!   
THE OLD WAY IS NOT WRONG!!!!

"But my child's teacher gave my kid a bad grade for not doing it their way."

     If your teacher is giving your child a bad grade for not following "the correct path" then the teacher is teaching the way they use to.  I use to fail math all the time growing up because my brain processes numbers in the "new way" but that was "unacceptable".  "A 7 is a 7. Not a (3+4) or a (6+1) or a (2+5)."<-- read that in a 60 year old math teachers voice.   When we all know that it is Technically the same.  For instance a 5 year old counting on their fingers can be considered "new math"  counting to 7 is (1+1+1+1+1+1+1)

     Common core math only ADDs new ways of doing the same math.  More paths to the same answer equals more fun in math as well as being able to find a better path for each person.  Some people run to a destination while others would rather swim (or fly or dance)
 
     So if you feel more comfortable doing it the "old way" keep doing it that way.  If a teacher fails you or your child for not doing it right.  Send said teacher to this blog.  They can learn what common core is.  An guideline of how to add more creative approaches to math.  Some people are linear thinkers (old way) while other people are creative thinkers (new ways).

     If the understanding of how numbers flow and fit together is there.  Then the answer should come out correct. (with the occasional human error here and there.)  As long as you are showing your process as to how you got to the correct answer then as far as "common core math" is concerned CONGRATULATIONS your a math wizard.

Common Core Math; Subtraction

      As I started with in the last entry I will start with a picture that has floated around the internet of a child's math test being taken out of context.  Of course same as last time it is a single random question that is worded in a different way of the "norm".  This particular question has no explanation from said teacher before the parent posted it to the internet. (note the green question mark)  At first glance this whole question is just plane confusing.  Somehow the answer is suppose to be 111 but "Jack" somehow managed to come to the conclusion of 121.

(Difficult worded Common Core Math Question)

      To start my explanation in a way that sheds some light on this whole issue is lets back track to the actual question that is asked which due to the fact that there is a bunch of random sideways C's grab your attention before you can notice anything else.  The actual question that is being said is;
      "Jack used the number line below to solve 427-315.  Find his error.  Then write a letter to jack telling him what he did right, and what he should do to fix his mistake."
      I will mark the whole "Find his error" part.  That means that the number line that everyone is confused about is wrong.  Then with all the parents numbers drawn on it causes it to look even more chaotic and confusing.  I will also have you notice all the lines given to explain things here.  Typically subtraction is taught in K-3rd grade depending on the schools and the age group.  For the sake of argument we will assume this question was for a 3rd grader on a 3rd grade reading and writing level.
        Is an 8 year old going to easily pick up the way this question is worded?  It has nothing to do with the math.  If this question had "He got this question incorrect explain what he did correct and how he could have fixed his mistake" instead of "Find his error then write a letter........"  The way this question is worded sounds like an adult is trying to sound impressive towards another person, instead of a teacher giving a question to an 8 year old.  So this issue is not the maths fault is that the teacher clearly needs to reword his English to be relevant to classes age group.
         The next part of course is the number of lines given for an explanation.  How many 8 year olds do you know that are going to need 15! lines to write what "Jack" did right/wrong.  At most 2-3 sentences.  So EVEN IF THEY WRITE BIG they will probably need 5-6 lines tops.
        Before I get right into this question I would also like to note the actual situation with the family trying to figure out this specific problem. I only dug thought the internet a little ways to learn that the father/writer of this pictures letter who I will refer to by his initials JS.  JS turns out, does have a Bachelor's in Engineering also has a son who has autism, attention disorders AND trouble with language arts.  Clearly the wording of this question was beyond him and the teacher should have taken that into account.  I am not an English teacher so even I had to read this question a couple times to fully understand what was going on with everything.  I feel the teacher could have used a much better way to convey the point of this question that best reflected the levels of his students.  This teacher is teaching math, not "comprehensive literature in math test questions".  My point is; This question is confusing due to the English.....not the math.
        For the sake of explaining why this problem has become so confusing I will have to show where both "Jack" and JS went wrong trying to figure out this problem with a line chart.  Even though they both made the same mistake but in slightly different ways.

Jack VS  JS

      Keeping In mind of course that I am not a Math artist.  This is Jack's work vs JS's work.
My clues in this picture are circles that I added in for the sake of understanding what they did to get where they ended up.
       So they both started at 427 obviously which is placed at the far right of the line chart.  Then the big humps have been indicated by both of them as 100's that are being removed via subtraction. (427, 327,227,127)
This is what they both did correctly.

Once they made it to the smaller humps they ended up getting a bit lost. Jack for instance was under the impression that each of the little humps were 1's thus (127,126,125,124,123,122,121) resulting in HIS answer of 121.  JS was under the impression that the humps were a (20) and 5(10's) so his chart went (127,107,97,87,77,67,57) resulting in HIS answer of 57.  If we were to go back and the the whole check your answer with addition thing (like we use to do in "regular" math) if we add all the numbers that were subtracted together what number do we end up with.
      I moved all the numbers I circled noting the amounts they were removing off to the right to add them all up.  If they were done correctly they should have added up to 316 right? because the math question was 427-316?  Jack only ended up subtracting 306, while JS ended up subtracting 370.
        The point of this question was to explain the idea that numbers can be split up and then subtracted separately the same way that it can be added together.
"But I haven't read your post about how to add separate numbers for addition"
 Then go and read that post first.  We typically learn addition first because we learn how to do things forwards first, then backwards. (examples; walking and counting)
        Anyway, moving on.  The number that they are attempting to separate is 316.  Jack and JS start out well by taking out the 100's first. which if they wrote out first would look a bit like this (100+100+100+16).  Once the 100's were removed on the line chart they were left with 16.  So instead of the question being 427-316,  It has now been changed to 127-16.  That one looks and sounds much easier then 2-large three digit numbers.
        That means the number that needs to be broken down now is 16.  There are lots of ways to break this number down.  The way Jack started was to separate them into all 1's but then he would have needed 16 little humps instead of only 6.  While JS's first hump was 20.  So clearly misunderstanding the fundamentals of how numbers work he somehow separated 16 into (20+10+10+10+10+10)<--this does not equal 16.
      "But I still don't understand.  How can I break 16 up in an easy way that fits into 6 humps!?"  Well let tell you a secret.  You don't have to!  Jack our imaginary friend,  made his chart wrong and then JS tried to force it to work.  The question says that Jack used the number line, basically he made the number line. If he wanted to he could have drawn 316 tiny little humps, or 2 humps of 158. It doesn't matter. "Common Core Math" or as it should be called "lets break up numbers so that they are easier for YOUR brain to process them without pretending they aren't real."

AND NOW FOR YOUR OWN AMAZEMENT! 3 WAYS TO SOLVE THIS PROBLEM USING COMMON CORE MATH!

(Actual Common Core Math)

Please excuse my sloppy handwriting.  Remember I am teaching math here not handwriting.

The first way is with the line chart.  It separates the 3(100's) same as before but then it removes a (10) and then 6 (1's).  Look you can already see a pattern.  

3(100's)+(10)+6(1's)=

You guessed it! 316!

So as you go back on the chart you find yourself counting down the hundreds,

(427,327,227,127)

Then by a 10,

(127-->117)

Then finally by ones.

(117,116,115,114, 113,112,111)

Tada! your final result turns out to be (111) which is in fact the correct answer.

"But I don't want to draw silly little line charts all over my paper every time I want to subtract stuff."

      Well I have good news!  The other two ways in this picture don't use weird charts.  Also, same as addition, if you practice doing math this way you will be able to see 427-316= and your brain will basically automatically spit the numbers. result will be, to answer this in your head.

        In the middle section I have a fun way to write out your subtraction problems where you can do math in a very similar was as you would for instance play Bingo.  You break all the numbers up into 1's, 10's and 100's. I kept them in ( )'s to help keep track of what is where.  Then the Bingo game begins!  Think of your first number as your Bingo board with your second number being the numbers called.  100 gets called once you cross them out in this case it gets called 3 times.  Result I crossed out 3 (100's) from my "Bingo" board, making sure I cross out the numbers from the ones called so I don't "call" them again.  Then the 10's followed by the 1's.  After I "called" all my numbers my "Bingo board" is left with (100+10+1)

Answer= (111)

        The bottom way is pretty much the same as the line chart.  But for those of us that can't draw straight lines very well.  We can just write it all out.  I calculated this the opposite way as before to show that it can flow either way.  427 is your starting number so I broke apart the 316 into 100's, 10's, a 5 and a 1.

"But can't I just stack them all up the normal way now and do the math?"  Yes, Yes you can. But lets test this out first.

      So This time I started with the lower numbers. (aka 1) we can subtract 327-1 right?  That makes 326.  That makes the next line (426-100-100-100-10-5=).  Our next number is a 5. So 326-5 still pretty easy.  This leaves our line as (421-100-100-100-10=)  Now that it's less complicated some of us are already doing this math in our head.

Now for the 10.  421-10=411.  So now with our line of math being (411-100-100-100=).  Some of us would just stick the 100's back together to subtract 411-300.  Which is SUPER easy.  However for the sake of math fun I removed the hundreds 1 by 1. 411-->311-->211-->111.

Answer = (111)

         Now to make this even easier to follow. MONEY!  If instead of 427-316 it was $4.27-$3.16.  <-- A lot of us can do this really quick in our heads using "Common Core Math".  We see these numbers separated automatically in our minds by money values. Dollars, Quarters, Dimes, Nickels and pennies.  Also when it's on paper we don't need to worry about going into a store to get change for a quarter because this money is imaginary. We know that a quarter is $.25 no matter how you change it. (2 dimes and a nickel, 1 dime and 15 pennies, etc.)  So $4.27-$3.16 = $1.11.  (Or if you feel like you want to pretend your carrying lots of money on you) $427-$316. It is still easy to figure out you'd have $111.  

        With the line chart method you can consider it like getting your paycheck then paying off bills.  You got paid $427.  You then had your cable bill of $100. Followed by your utilities of $100.  Then your cellphone bill for $100.  Put some gas in your hybrid $10.  Then your 6 kids wanted their $1 allowance.  Your left with $111.  Which if your luck is anything like mine, it will probably go to somewhere else anyway leaving you with barely anything for the week.

"Okay Okay, This makes sense now but what about when your numbers aren't so easy and you need to borrow numbers from other numbers?  Like when you subtract 47-19.  You would need to borrow from the 4(in 47) to turn the 7 into a 17."

      Well if you think about it, when you are "borrowing" from the two.  You are actually doing "common core math"  you separated the 40 (aka the 4) the your spiting it into (30+10), and then moving the 10 to the 7 and adding it to 17.   So now your technically subtracting (17-9) and (30-10).  You are still breaking down the problem but your doing it in the way that you have always been told it HAS to be done.  Not everyone wants to subtract 17-9.  But of course after you subtract those two parts aka the "1's place" and the "10's place"  You come up with 28 as your answer.  <-- This way is NOT wrong.

(This is why there is no difference between "Common Core math" and "normal math")

        If you wanted to do this "Common Core" style. Then I will show you.

Granted I used the my little "Bingo" method here as an example but you could have easily done this with the line chart or even just writing it out and then subtracting each piece of the broken up 19 from the 47.
      For the way I did it here, I started by breaking up all the numbers into 10's,5's and 1's.  However I quickly realized that during my Bingo game part I wasn't going to have the correct spaces on my "board".  So instead of trying to "borrowing" aka subtracting then adding before subtracting again. I took one of my extra 10's that I wasn't going to need and I spit the number down even more so.  So that 10 was broken up into (5+1+1+1+1+1).  Then I could start my bingo game.  (I could have also started it before, paused to break up the 10 and then continued). Anyway I removed a "called" 10.  Followed by a 5.  Then the 4(1's).  Which left me with (10+10+5+1+1+1).  Which can be slowly added back together like I did in the picture. Or quickly added together in your head by using the simplicity of 10's, 5's and 1's.
The answer is still 28.

        Of course you can keep in mind that these are just some of the ways to do subtraction.  There are still the count up methods.  aka 9-6=  Instead of down from 9 for 6 places. You can count up from 6 (7,8,9) for 3 places and 3 is your answer.  Which can easily be done with larger numbers.  325-38.  Start at 38 add to nearest 10. (add 2) makes 40, add to nearest 100. (add 60) makes it 100, add to the closest 100 without going over. (add 200) getting you too 300.  Then finally add to the first number. (add 25). Add up the numbers in ( )
(2+60+200+25)=287

        So that is all I am going to explain for this post.  Next time I will be talking about multiplication.  That will be fun.  However I should warn you that sometimes the stuff on the internet is not true.

Unexplained count up method

        This last picture right here ------------------------>
This my friends is technically common core. But a not good explanation on why it works.
They start at 12. To get it to the 15 (5s are easy) they add 3.  Then to get 15 to 20 (10s are easy) they add 5.
then to get 20 to 30 (which is as close to 32 as they can get with a 10 which is easy) they add 10.  Then to get 30 to 32 they add 2 (obviously) then you add up those places you move up (3+5+10+2=20 aka answer)

For me; I would have counted down because its (30-10) + (2-2)

However for the count up method;
I would start at 12. and add 8 (to get to the 10s place)
then Id be at 20 already. Add 10 (to get to 30) then add 2 to get to 32.
addition part then becomes 10 + 8 + 2 = 20
(easy because 8+2 is 10.... and 10+10 is 20)

Common Core Math; Addition

          Now a bunch of people have see this weird picture of a child's math test and have seen the the "OH man common core makes no sense." But what if i was to tell you that this question and its answer does in fact make sense.  Granted taken out of context where you only see this question with no explanation mixed with the limited space the teacher had to write the explanation as to the answer it does seem pretty stupid.  However the question doesn't ask "What is the answer to 8+5." It is asking "how do you make 10 when adding the numbers 8+5."

But if you have read my previous blog about how "common core math" really just teaches you to break up numbers it makes sense.

8+5= (for us adults its easier to do it in our head) BOOM answers 13.
But what about the children just learning?
They think we are just magic guessers of the answer.
The old math VS the new math. There is really no difference to it if you really think about how we teach addition.
example of the old way. We teach kids at first to count on their fingers or dots on paper.
They hold the number 8 in their minds and count up one at a time adding 5 more to it.


On paper that would look like this 8+(1+1+1+1+1)=13.
But silly thing is during this process they make a 10 appear in their heads. 8+1+1=10.
Seeing such stuff on paper makes it easier to understand how numbers flow together.  8+5=13.  Not because we are force to memorize it but because it teaches our minds to factor all the numbers into a different form to better understand how they connect.
As you get better at numbers not everyone needs to break numbers into 1's. They can be broken up into anything.

example;
8+5 <-- original math problem.
8+(2+3) <-- Same as original but slightly broken down. And if you did it this way would still have the same answer.
You can then add (8+2) to make 10.  leaving your new math problem..
10+3 <-- Same as original only shifted slightly to make it easier to do.
Even though 8+5 is the same as 10+3. If you give kids flash cards of math problems they will add 10+3 much faster.

        But why am I telling you that this makes it easier.  Well thing about how we first teach our children about numbers.  We teach them to count. We start by 1's and even toddlers can grasp it to a specific amount.  Then by 2's and 5's and 10's.  1's, 2's and 5's all count into 10's that makes this group of numbers very easy to keep track of mentally.
      It is only when we are given weird numbers that math starts to slow down and seem more complicated to kids. 3's, 4's, 6's, 7's, 8's and 9's.  It takes longer for these numbers to loop back into an easily tracked number.
3's don't loop to a 10's group until they add up to 30.
4's -->20.  6's -->30, 7's -->70, 8's -->40, 9's -->90.
If you need proof of this idea of how we understand numbers look at money.
We have $1,$5,$10,$20,$50,$100.etc
Same with our coin currency.
pennies(.01),nickles(.05),dimes(.10).  Even quarters(.25) are easy numbers to follow because the .20 and the .05 are easily added into .50 so 2 sets of 2 quarters is $1(1.00).

There are lots of ways to break down the question of 8+5 into a way that can be easier for everyone.
you can separate a 5 out of the 8 to add 5's together easily into 10.
8+5=
(3+5)+5=
On paper this seems pretty stupid for an adult but we do stuff like this all the time in our daily lives.  If you ever use cash to pay for stuff chances are you do it frequently.

You walk into a store and you have $8 in your right pocket and $5 in your left pocket.  You know you have $13 in your pocket.  You walk up to the counter and buy a $10 lottery ticket. 

      When the cashier asks for $10 to pay for said lotto ticket you don't look at him and say "sorry I only have $5 or $8 or $13" and you don't typically dump all of your money on the counter and then have the cashier hand you back $3.
      I mean you CAN do these things but the cashier will typically give you a "what in the world is up with this person" look and/or assume you don't understand how money works.
You take $5 from one pocket and $5 from the other to make $10 and pay for your ticket.  Leaving $3 left in your one pocket.

      However you just bought a lottery ticket so there is a VERY large chance you just wasted $10. Might I suggest buying something else next time?  But congrats you just did common core math OR as I like to call it, MATH.

NOW FOR THE BIG STUFF! (because everything in the world is not so small.)
      Perhaps your math problem is a bit bigger now.
243+428=
      Before anyone starts hyperventilating about doing this any other way then the "normal" way.  The new way is pretty much the same.
      I will start with the "old" way to solve the problem.



     Basically this way to add up numbers means stacking them up and drawing lines separating the 1's 10's and 100's places. (either by actually drawing them or just mentally placing lines down the numbers.)
     From there you can pretend there aren't any 10's or 100's. It is now just 3 separate little math problems that are still technically stuck together.
      In this specific problem there is only one issue with doing all the different parts separately and that is the 8+3 part.  In this part of it we teach kids they need to just already know that 8+3=11; but then they need to break the answer they just figured out back up into 10+1 so they can move (carry) the 10(which gets its 0 cut off and moved to the 10's row) and the 1 stays in the 1's row.  Then now they have to add 3 numbers together in their heads.  In this math problem its easy but sometimes its not so easy for mental math.
The 10's row is (1+2+4=7) pretty easy for us adults that add 2 numbers then add the 3rd to the answer.  If you were 4-6 would that be so easy of a process?  or if instead the numbers were say 1+7+9=17 only to do more carrying over to the next line of numbers?
      So basically this way we teach kids that the 4 of the top number is a 4 in the hundreds spot because 4s are easier then 400.  When in reality its still 400. so when your adding the 4 and the 2 your actually adding 400+200.

And now we move forward to the dreaded COMMON CORE MATH!. DUN DUN DUN.

      I know it looks terrifying and like it has WAAAAY to much effort to figure out the same thing,  but like any good relationship lets talk it out first, Okay?

  243+428 looks kind of big for those of us who can't just automatically piece numbers together.  However like any other common core question lets break up the easy parts first.
      At first I broke down both of the two numbers.
(100+100+10+10+10+10+3)=243  +  (100+100+100+100+10+10+8) =428
      Still following? good.  So we went over numbers that are easy peasy lemon squeezy. So we are just going to leave all the 100's and 10's where they are.  We will get back to them later.  That literally only leaves us (3+8).
      "but how is this any different then the other way then?" you may say.
  Exactly it's very similar.
      Doing the same process as our little 5+8 question earlier we know that if we take 3+8 we can find a 10 to get out of our way same as all the other 10's.
3+8=
(1+2)+8=
        Then by adding the 2 into the 8 you make 10. TADA! now you only have 1 in our annoying number section. Instead of 3 and 8.
        Then you throw the numbers back up.  Thus the circled 10 has an arrow pointed back up to the 8 showing that the 8 was separated from that group. shifted with the 3 and then put back as a 10.
        Now if we write it all down with it being slightly adjusted for easier comprehention.
(100+100+10+10+10+10+1)+(100+100+100+100+10+10+10)=
        Then we add ALL the 100's and 10's back together as easily as counting 1,2,3.
600+70+1=671 -BOOM!- we just did some math!
      "But wait I don't understand how this makes any sense for practical use in the real world."  
       Funny you should ask that.  On paper this process seems very silly and drawn out and otherwise a large waste of time.  But what if we do this ALL the time.  In fact we do.

      Okay now before you make a joke about not having any money.  I find that the less money we have the more likely we do this process. 
*RIDDLE TIME!*
What do we do more with money when we have less of it?
.
..
...
..
.
COUNT IT!
      What if instead of the numbers being 243+428 the numbers were converted into money.
"Yea cause I totally carry around $671 right?"  Nope but close.
$2.43 + $4.28
So instead of 100's they are actually $1(aka 100 pennies)
and instead of 10's they are Dimes(aka 10 pennies)
So $2.43 is more likely 2($1), 4(Dimes) and 3(pennies)
And $4.28 is more likely 4($1), 2(Dimes) and 8(pennies)
So if you mix it all up on the table you unintentionally seperated everything the way we did earlier that seemed like a waste of effort.

Anyway moving on...
      So when I count change.  In this case the pennies I typically group them. so instead of a pile of a weird number 11 pennies but adding all the pennies together I would have grouped them into a group of 10 and then kept the other one separate. Thus creating the 10+1.  Then when I add up all the $1's i would have $6's in ones.  $0.70 in dimes (and the group of 10 pennies).   Then one remaining penny.  Thus giving me $6.71

      We end up doing all this entirely without thinking about it when we get good at math.  However Common core math is not "new math" that makes the old one wrong.  It is just an easier way of explaining it to kids/people who just aren't able to do it in their heads yet.  After learning it this way it because more of an understanding on WHY the numbers become what they do with adding/subtracting/multiplying/dividing instead of just memorizing them as they are with flash cards because everyone told you that 8+5 is 13.