And keep in mind that I do talk to myself as if I am someone else. The does include arguing with myself.
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| 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.
~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.
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.
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| OLD METHOD NEW METHOD |
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;
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| 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.
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;
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| 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!!!
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