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.

Warning: I find math fun.

Anyone having trouble understanding how "common core" math works?
 I am going to start posting explanations on how it works. In a way that is not out of context(like random pictures on the internet) It actually doesn't change the other way it just explains how numbers work to better comprehend the "original math"




The "normal" math makes numbers seem set in stone like letters.
example; (A=A) (452=452)

But numbers aren't set in stone. which is why we unintentionally do better math when we think of money.
example;
(452)=(400+50+2)        OR (100+100+100+100+25+25+1+1) etc...
(4.52)=(4.00+.50+.02) OR (1.00+1.00+1.00+1.00+.25+.25+.01+.01) etc...

Math problems can be broken down into easier to comprehended versions and then stuck back together like a puzzle. Making math easier to do in your head later on when you can skip all the steps just because you happen to know how the numbers fit together.

Like memorizing multiplication tables on a much larger scale but without the effort of expecting everyone to memorize everything.