Math question. Why does this work?

[Alice]

I was working on mental math with my son this morning. We use Singapore. He's naturally pretty good at mental math so I first asked him how he would do the following:

 

45-8

 

His answer was that he thinks 8-5 is 3 and then 40-3 is 37.

 

Why does this work? He doesn't know why he does it this way, it just seems natural to him. (This isn't the way Singapore teaches. They would say to think that 45-8 is the same as 30 + 15 -8. )

 

This is one of those elementary math things that seem simple but that I can't figure out why it works and I end up feeling really stupid. :tongue_smilie: It reminds me of reading the Liping Ma book. I had new appreciation for the fact that I'm good enough at the mechanics of math but never really understood the why very well.

 

I'm ok with him doing it his own way but I'd like to be able to tell him why it works.

[TXBeth]

We are only in Singapore 1A, but they are starting to teach things this way. The "8-5" part basically represents breaking 8 up into a number bond of 3 & 5. The 5 cancels out the 5 ones in 45, and then you have to subtract the 3 from the remaining 40. So basically, you subtract 5 from 8 to see how many of the 8 still need to be subtracted from the 40 when you run out of ones.

 

I don't know if I'm explaining well, but it makes sense in my head. :)

[Guest RecumbentHeart]

We are only in Singapore 1A, but they are starting to teach things this way. The "8-5" part basically represents breaking 8 up into a number bond of 3 & 5. The 5 cancels out the 5 ones in 45, and then you have to subtract the 3 from the remaining 40. So basically, you subtract 5 from 8 to see how many of the 8 still need to be subtracted from the 40 when you run out of ones.

 

I don't know if I'm explaining well, but it makes sense in my head. :)

 

 

I know what you're saying. :)

 

Without really thinking about it, probably, he's subtracting 5 from both to make a simpler equation. It's great that he knows the parts of 8 so well that he does this naturally.

[Alice]

Hmmm....that does make sense. Thanks. Funny thing is I did teach him number bonds but never really thought of that here. Seems he remembers them better than I do. :)

 

Is there a fancy name for the principal of why this works? Or is it just playing with numbers?

[Spy Car]

I was working on mental math with my son this morning. We use Singapore. He's naturally pretty good at mental math so I first asked him how he would do the following:

 

45-8

 

His answer was that he thinks 8-5 is 3 and then 40-3 is 37.

 

Why does this work? He doesn't know why he does it this way, it just seems natural to him. (This isn't the way Singapore teaches. They would say to think that 45-8 is the same as 30 + 15 -8. )

 

This is one of those elementary math things that seem simple but that I can't figure out why it works and I end up feeling really stupid. :tongue_smilie: It reminds me of reading the Liping Ma book. I had new appreciation for the fact that I'm good enough at the mechanics of math but never really understood the why very well.

 

I'm ok with him doing it his own way but I'd like to be able to tell him why it works.

 

The "difference" between 8 and 5 is three. In the case where the place value of the minuend is 5 and the subtrahend is 8 that "difference" is expressed as -3 (negative three). Negative three and forty make 37.

 

Bill

[Guest RecumbentHeart]

Let's make it really complex .. just for fun!

 

45 - 8 = x

(40+5) - (3 + 5) = x

1(40 + 5) - 1(3 + 5) = x

40 + 5 - 3 - 5 = x

40 - 3 = x

37 = x

 

And he did it without much thought. :lol:

[lmrich]

My dd does the same thing. I am glad that she can make it work for her. I love listening to her think aloud. I am not skilled at mental math and always show my work!

[Spy Car]

Let's make it really complex .. just for fun!

 

45 - 8 = x

(40+5) - (3 + 5) = x

1(40 + 5) - 1(3 + 5) = x

40 + 5 - 3 - 5 = x

40 - 3 = x

37 = x

 

And he did it without much thought. :lol:

 

I wanna break out some C Rods and make it really easy :D

 

Bill

[Alice]

I wanna break out some C Rods and make it really easy :D

 

Bill

 

Anyone who throws words like minuend into an answer isn't looking for easy. :D

 

Thanks everyone for the help! I knew I'd get a fast answer here.

[Guest RecumbentHeart]

Anyone who throws words like minuend into an answer isn't looking for easy. :D

 

Thanks everyone for the help! I knew I'd get a fast answer here.

 

 

The rods are the universal language so he can communicate with the rest of us.

 

I have to chuckle because I can't help but imagine: "Mom, go get your c. rods and I'll show you ..." :lol:

[Spy Car]

Anyone who throws words like minuend into an answer isn't looking for easy. :D

 

Thanks everyone for the help! I knew I'd get a fast answer here.

 

How about "the top" number? :lol:

 

But seriously, think of a number bond, or of two C Rods. The "difference" between 8 and 5 is 3.

 

If we want to take a total of 8 away and we remove one "part" that is 5, we would still need to remove another "part" of 3 to remove a total of 8.

 

Bill

[boscopup]

I thought Singapore taught subtracting to 10s as one of the strategies? Here is how Math Mammoth breaks it down when teaching it:

 

45 - 8

.......|.\

(45-5)-3

40-3

37

 

Hopefully that's understandable. I think your son's method was a good one, and it's what I would have used. You subtract down to the 10, then what's left from the single digit gets subtracted from that result.

[stripe]

45-8

 

His answer was that he thinks 8-5 is 3 and then 40-3 is 37.

 

Why does this work? He doesn't know why he does it this way, it just seems natural to him. (This isn't the way Singapore teaches. They would say to think that 45-8 is the same as 30 + 15 -8. )

 

I interpret what he's doing like this

45-8 --> 40+5 - 8

since 8 = 5+3

45 -8 = 40+5-5 -3,

which could also be imagined as 40 +5 - 5 -3 = 40-3

so 40 - 3

 

This seems like MEP style (maybe MM too?).

 

**ETA I combined a few thoughts, it came out wrong the first time!

Edited May 2, 2011 by stripe

[Spy Car]

The main mental math way Singapore would teach 45-8 would be to think in terms of tens. So 45 would be the same as 35 and 10.

 

35+10-8= 35+2

 

35+2=37

 

There are obviously other mental math strategies, and Singapore generally encourages knowing and implementing several. But this would be typical strategy.

 

Bill

[boscopup]

Thanks, Bill. That makes sense. The 30+15 way seems not as easy to me, as it uses 15-8 instead of a more simple fact like 5+3 or 10-8. Obviously it works, and there's nothing wrong with that method, but it's not one I'd likely use for that particular problem.

 

I naturally make 10s, though I use a variety of strategies.

[nmoira]

I interpret what he's doing like this

45-8 --> 40+5 - 8

40 -8 = 40-5 -3,

which could also be imagined as 40 +5 - 5 -3 = 40-3

so 40 - 3

 

This seems like MEP style (maybe MM too?).

Is does sound like MEP. DD the Younger would say, "Five to forty and take away three more to thirty-seven."

[Spy Car]

Thanks, Bill. That makes sense. The 30+15 way seems not as easy to me, as it uses 15-8 instead of a more simple fact like 5+3 or 10-8. Obviously it works, and there's nothing wrong with that method, but it's not one I'd likely use for that particular problem.

 

I naturally make 10s, though I use a variety of strategies.

 

This method has become very natural to me.

 

There is another method that we have learned to employ that I often find even easier, one we call "bumping up" (although that is not an official name). This would involve adding a sufficient value to the subtrahend to "bump it up" to the nearest Ten (or Hundred, Thousand, etc). And adding the same amount to the minuend.

 

So instead of dealing with 45-8 we add "2" to both sides of the equations and get:

 

47-10=37

 

Mathematically valid and often the fastest way to calculate (at least for me.)

 

Bill

[leeannpal]

My daughter does mental math very well, and she would have done the problem in this way. Leaving 45 as 45, she would have rounded 8 up to 10, subtracted 10 from 45 and added the 2 she got from rounding up the answer

 

45-10=35

35+2=37

[tristangrace]

Thank you! My daughter did the same thing about this time last year, and I felt like a FOOL not being able to understand what the heck she was doing . . . and why it kept working. :)

 

I was one of those students who did fine in math ONLY because I could memorize the processes--but actual creative thought about how math works gets me incredibly flustered. That incident is what pushed me to look for math programs that emphasized concept as well as process.

[crazyforlatin]

I love that you all seem so excited about solving this problem in different ways! I'm like this now as well, but you wouldn't have thought so when I was in school.

[boscopup]

This method has become very natural to me.

 

There is another method that we have learned to employ that I often find even easier, one we call "bumping up" (although that is not an official name). This would involve adding a sufficient value to the subtrahend to "bump it up" to the nearest Ten (or Hundred, Thousand, etc). And adding the same amount to the minuend.

 

So instead of dealing with 45-8 we add "2" to both sides of the equations and get:

 

47-10=37

 

Mathematically valid and often the fastest way to calculate (at least for me.)

 

Bill

 

Yes, I do that a lot with hundreds and thousands. It's amazing how many different ways you can add or subtract in your head. :D

 

Now my son will come up with his own weird ways that don't make much sense to me (as to why he'd do it that way), but hey, whatever works, right? For example, one day we were doing 8 + 7 (before we worked on the facts to 15), and he said "Oh, that's 12 + 3, so 15!" Not exactly a technique I would have used, but he came up with the answer immediately, so it obviously worked for him. :) My sister said she would do the same thing (and she was a physics major and is very good at math). My son is soooooo much like my sister, it's scary. :lol:

[Floyd]

When we were doing "subtract single digit from double digit with renaming" (which 45-8 is), we taught her four different methods and gave her homework where she had to fill out a grid using all four methods:

 

a) Split out one of the 10s, subtract from that, then add it back in: 35+2=37. (we call this "shiny dime + grimy pennies"; only break up the dime if you need to).

 

b) Similar to above, split the 45 into 40 and 5, subtract 8 from the 40, then add back in the 5: 32+5=37 (We call this "separate and recombine 10s and 1s")

 

c) Split the 45 into 30 and 15, subtract 8 from the 15, then add back in the 30: 30+7=37 (SM calls this "one less ten", and is the same as "borrowing" as most of us learned way back when).

 

d) Subtract the 8 in two chunks: 5 to get down to 40, then the remaining 3 takes you to 37. We call this "train line". Another train line method is add "5 to get from 45 down to 40", "2 to get from 8 up to 10", then "30 to get from 10 to 40". This is my favorite method, especially for subtracting larger numbers.

Edited May 2, 2011 by Floyd

[JMDRAD]

I love these mathy threads!! I always learn something from you guys.:D

[8filltheheart]

One thing that always surprises me when people make comments like "well this is the way such and so teaches it." I guess I don't understand NOT having these types of conversations when teaching math. My kids would look at 45-8 and would probably do it mentally 45-10+2 b/c we always discussed mental math with 8s as 10-2.

[stripe]

One thing that always surprises me when people make comments like "well this is the way such and so teaches it." I guess I don't understand NOT having these types of conversations when teaching math. My kids would look at 45-8 and would probably do it mentally 45-10+2 b/c we always discussed mental math with 8s as 10-2.

I have always made up my own ways too. It just reminded me of some of the techniques introduced in MEP that was different from how I learned, but seemed clever to me.

[Spy Car]

One thing that always surprises me when people make comments like "well this is the way such and so teaches it." I guess I don't understand NOT having these types of conversations when teaching math. My kids would look at 45-8 and would probably do it mentally 45-10+2 b/c we always discussed mental math with 8s as 10-2.

 

I'm surprised that you're surprised :D

 

The math programs I was raised on taught column addition and subtraction, and procedures like "borrowing and carrying" and no mental math techniques. Any "strategies" were learned ad hoc by individual students without the aid of the curriculum or instructors.

 

Seems to me some programs still limit instruction to the "standard algorithm." So I don't understand quite why you're surprised.

 

Bill

[Alice]

Thank you! My daughter did the same thing about this time last year, and I felt like a FOOL not being able to understand what the heck she was doing . . . and why it kept working. :)

 

I was one of those students who did fine in math ONLY because I could memorize the processes--but actual creative thought about how math works gets me incredibly flustered. That incident is what pushed me to look for math programs that emphasized concept as well as process.

 

Sure, always glad when my own ignorance can help somone else. :)

 

I was the same way, I went very far in Math in school and did fine as far as grades but I never really had a mathy brain.

 

One thing that always surprises me when people make comments like "well this is the way such and so teaches it." I guess I don't understand NOT having these types of conversations when teaching math. My kids would look at 45-8 and would probably do it mentally 45-10+2 b/c we always discussed mental math with 8s as 10-2.

 

I think one thing I've realized is that I've been too lax about actually teaching this kind of stuff because ds does just do it naturally. I'm fine with him just doing things his own way as long as it's valid and works but I also want to make sure I'm stretching him to think about things in other ways too. So far, Math has been easy for him and so it's been too easy for me to teach.

[Alice]

The math programs I was raised on taught column addition and subtraction, and procedures like "borrowing and carrying" and no mental math techniques. Any "strategies" were learned ad hoc by individual students without the aid of the curriculum or instructors.

 

 

 

Yes! I'll admit how I'd actually do this problem myself. One of two ways. One is that in my head I visualize a numberline and quickly just sort of count down from 45 to 37. That's not something I was every taught but whenever I think of numbers they are on this number line in my head. It even has specific places where it turns corners.

 

The other is that I visualize writing out the problem in a column and do it that way with the borrowing.

 

Neither is a particularly great way of doing mental math and it's something I've always been poor at. It's also one of many reasons why I like Singapore.

[boscopup]

I'm surprised that you're surprised :D

 

The math programs I was raised on taught column addition and subtraction, and procedures like "borrowing and carrying" and no mental math techniques. Any "strategies" were learned ad hoc by individual students without the aid of the curriculum or instructors.

 

Seems to me some programs still limit instruction to the "standard algorithm." So I don't understand quite why you're surprised.

 

Bill

 

:iagree:

 

I do the mental math strategies myself, but no one taught them to me. I was a "mathy" child and just figured them out. The same goes for my DH. Not everyone figures out these techniques though. Someone who struggles in math all through school may have never thought of such techniques, and now they're trying to give their child the math education they never had. Hard to explain mental math techniques if you can only do math by writing out the standard algorithm or using a calculator. Some people need the curriculum to teach such things because they either don't know how to explain it or they never learned it themselves. Nothing wrong with that! I prefer having the curriculum explain it, as I'm still getting used to how to explain a concept like that to a child.

[stripe]

Are those strategies books they sell at Singapore Math any good?