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| (Difficult worded Common Core Math Question) |
"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.
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| Jack VS JS |
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!
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| (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.
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.
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| 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)
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