Ideas for getting the subtraction concept

[bookbard]

My 8 yr old is still struggling with subtraction. Not the idea - he knows it's taking away, he can do word problems (he's doing year 3 beast academy). But once it gets beyond the basics, he still feels that he can move digits around happily to make it easier. He can agree that 23 is smaller than 39, so if you take 39 from 23 you end up in negative numbers .  . . but he'll still end up with 16 as the answer (if, say the sum was 123-39, he'll end up with 116). There's a gap between the language concept and the maths concept, somehow.

He's been tested and he's in the 99th percentile for language, and average for the rest. He's in school at the moment, which, as we're under lockdown, means I'm homeschooling him using school materials (which is super relaxed - nothing online, just a few worksheets). But it's a good opportunity for me to try and work on this concept.

[HomeAgain]

I would either use place value discs or base 10 blocks.

My extra 9 & 10yos have a tendency to groan because they know I won't give them the answer.  I'll make them defend what they wrote by doing it another way - either drawing it or using the blocks/place value discs.  Their job is to prove that they're right.  It takes a few times, but making them do it in a more tactile way helps strengthen the mental process and really grasp what is going on.

Guesthollow has free printable place value sheets and counters.  They work well if you print on cardstock.

[bookbard]

Yes, I've got blocks (cuisinaire) - I will dig them out again, I think. The thing is, he can 'do it' using a number line or drawing up blocks and crossing them out. I just wonder how he gets from there to seeing the algorithm and being able to conceptualise it, without drawing it all up. 

[HomeAgain]

4 minutes ago, bookbard said:

Yes, I've got blocks (cuisinaire) - I will dig them out again, I think. The thing is, he can 'do it' using a number line or drawing up blocks and crossing them out. I just wonder how he gets from there to seeing the algorithm and being able to conceptualise it, without drawing it all up. 

My extras like two different ways, depending on the kid, so I probably should have written that out above.

1st way: set out the discs or blocks for the total amount.  I can ask "Okay, how much are you taking away/how far have they gone already? or whatever the word problem is.  Then after telling me, he must remove that amount, working from right to left (because we're practicing composing/decomposing tens, too).  What is left is the answer.

2nd way: addition.  Write the two numbers with a space between them to draw a line.  Then he makes "hops", starting with the number taken away and adding units, tens, hundreds in that order.  So with your problem, it would look like:
39___+4_____+80______123.
                43               123

It's a quick way to prove that the problem is right, and works on a different skill (eventually using bar models for word problems).  My subtraction-shy kid will often do it this way because it's quicker for him and he feels more confident, We work on hopping backward as well, chunking it out mentally with -23 to 100, -60 to 40, and -1 to 39.

I think it's just a development thing.  There's a lot to work on in early math because it's the foundation for everything else.  A kid who can do the four operations easily and understand what is going on will do just fine in later math.  Eventually, one day, it clicks and they have a good mental model to draw from.

[purpleowl]

You mentioned that he understands you end up in negative numbers - does he have an understanding of negative numbers? My son would look at 123-39 and say that's one in the hundreds, negative one in the tens, and negative six in the ones. Then he says okay, one hundred and negative one tens is ninety, so that's "ninety-negative six," and that's eighty-four. Wondering if that approach would work for your son as well? I know it's an odd way to do it, lol.

[HomeAgain]

3 minutes ago, purpleowl said:

You mentioned that he understands you end up in negative numbers - does he have an understanding of negative numbers? My son would look at 123-39 and say that's one in the hundreds, negative one in the tens, and negative six in the ones. Then he says okay, one hundred and negative one tens is ninety, so that's "ninety-negative six," and that's eighty-four. Wondering if that approach would work for your son as well? I know it's an odd way to do it, lol.

This brings up a good point. 
I made a subtraction cheat sheet for my kid, with three different ways of subtracting.  One of the ways was crossing out the same on the top and the bottom numbers so that you were left with the most basic problem.  I'll see if I can find it on this computer and add it to this post.

ETA: found it, but it requires another piece of understanding, that the difference between two numbers is a fixed length, where you can add or subtract the same amount to both numbers and you are just moving the length, not changing the length of space between.

Edited August 13, 2021 by HomeAgain

[Not_a_Number]

Does he always think of subtraction as taking away a number? What’s his visual for subtraction?

I’m curious if this is a place value issue or a subtraction issue. Could be either.

Edited August 13, 2021 by Not_a_Number

[bookbard]

12 hours ago, Not_a_Number said:

Does he always think of subtraction as taking away a number? What’s his visual for subtraction?

 

I got him to do a few today on paper, no distractions, no worksheet - just paper and the sum.

He could do them, by drawing out the blocks. I think he just isn't invested in getting the right answer - his 'method' of changing around the numbers gets him finished, whether it's right or not. So I'm wondering whether at school there's less checking of correct answers and now I'm watching what he's doing, he's not getting away with it. I will continue giving him just paper and sums to do over the week, without the fancy worksheets as distractions, and see how that goes. 

[Not_a_Number]

41 minutes ago, bookbard said:

I got him to do a few today on paper, no distractions, no worksheet - just paper and the sum.

He could do them, by drawing out the blocks. I think he just isn't invested in getting the right answer - his 'method' of changing around the numbers gets him finished, whether it's right or not. So I'm wondering whether at school there's less checking of correct answers and now I'm watching what he's doing, he's not getting away with it. I will continue giving him just paper and sums to do over the week, without the fancy worksheets as distractions, and see how that goes. 

I wouldn't be surprised if the school doesn't check. When DD9 went to kindergarten, she regressed on some of her letters, since no one ever checked them.

However, I would probably facilitate this by letting him have blocks or some other place value manipulative. My experience is that the more experience a kid has with a model, the easier things are, and the less effort it takes. 

[PeterPan]

12 hours ago, bookbard said:

his 'method' of changing around the numbers gets him finished,

Have you shown him part-whole circles? He's very bright realizing the relationships between the numbers, so he just needs to realize WHAT he is discovering/catching onto. Then he can separate out the steps, seeing the difference as one of the circles in the part-whole diagram and the sign considered after that.

Why is he doing digit math written anyway? It should all just be done mentally. You back up and give him simpler 2 digit addition and then subtraction problems for him to work with.

Ronit Bird has a hand game Positive/Negative Turnovers which you can get instructions for using her free Card Games book download (ibooks). http://www.ronitbird.com/

[bookbard]

Actually, I agree that it should be done mentally. I think they've made a big jump at school and I need to go back. And, I've just discovered, all the other parents in his class are talking about this on fb. They're saying they're giving up on all the models, number lines etc and going straight to the algorithm they were taught themselves. This is going to be an issue with lockdown I guess - parents having to teach new maths methods when they don't know them themselves. 

[Not_a_Number]

1 minute ago, bookbard said:

Actually, I agree that it should be done mentally.

I don't think it matters a ton, although I do find that kids who are comfortable with the concepts can eventually do it mentally... at least if they have enough working memory to do it, they can. Some kids have trouble storing enough stuff in their head at one time, though. 

 

1 minute ago, bookbard said:

I think they've made a big jump at school and I need to go back. And, I've just discovered, all the other parents in his class are talking about this on fb. They're saying they're giving up on all the models, number lines etc and going straight to the algorithm they were taught themselves. This is going to be an issue with lockdown I guess - parents having to teach new maths methods when they don't know them themselves. 

Oh, that's too bad 😕 . 

There aren't really any "new" methods here. There's place value, and that's it. The more experience he gets hands-on with place value, the easier this will be. (And for what it's worth, I do NOT like number lines for addition and subtraction. At all.) 

[bookbard]

I would like to see the evidence that teaching both number line addition/subtraction and the blocks - often on the same worksheet - is helpful. Surely it would be better to have one mental model?

[Not_a_Number]

4 minutes ago, bookbard said:

I would like to see the evidence that teaching both number line addition/subtraction and the blocks - often on the same worksheet - is helpful. Surely it would be better to have one mental model?

Oh, you will NOT get that evidence from me. Yes, please, ONE mental model. And my favorite model is NOT the number line. 

[HomeAgain]

6 minutes ago, bookbard said:

I would like to see the evidence that teaching both number line addition/subtraction and the blocks - often on the same worksheet - is helpful. Surely it would be better to have one mental model?

I'll say I think both are helpful.  I like a line with word problems that talk about distance, length, etc.  It's easier for a kid to visualize the information and what it's asking for way.

For standard problems I like block models, or ones that rely on place value more heavily.

[Not_a_Number]

5 minutes ago, HomeAgain said:

I'll say I think both are helpful.  I like a line with word problems that talk about distance, length, etc.  It's easier for a kid to visualize the information and what it's asking for way.

For standard problems I like block models, or ones that rely on place value more heavily.

I haven't found both to be helpful at once. I find that to do distances, you simply need to have a good visual that "subtracting" is "taking away." Then the distance questions simply make sense. 

But to make sense of the place value, it's good to focus on the place value. Trying to focus on both will just leave lots of kids confused. 

[goldenecho]

On 8/13/2021 at 2:47 AM, bookbard said:

 But once it gets beyond the basics, he still feels that he can move digits around happily to make it easier. He can agree that 23 is smaller than 39, so if you take 39 from 23 you end up in negative numbers .  . . but he'll still end up with 16 as the answer (if, say the sum was 123-39, he'll end up with 116).

I'm thinking using 10 frames might help, and going over place value more.   Math U See has a really good way of describing place value, and it's in one of their sample videos, so maybe watching that with him would help. 

[Homeschooler_CH]

I’m listening in too. My 8yr old is also having trouble with Subtraction. He doesn’t seems to be strong in place value so that is my plan with him for start of the year spend as much time needed on it.

 

My elder child was stuck too at the same spot but he had pretty strong hold of place value. He said his brain needs to think more if he does regrouping in traditional way. I’ve stumbled on below while teaching him and this is how he prefers to do it.

Steps are more but he says it’s pretty quick way to do it.

https://www.mathsisfun.com/numbers/subtraction-quick.html

 

Edited August 15, 2021 by Homeschooler_CH

[knitgrl]

It's nice to know other kids have an issue with this as well. My son can do easy subtraction, but having to borrow trips him up. He would say that 33-19=26, because 9-3=6 and 3-1=2. So I guess we need to work on place value some more. It's challenging because he almost completely shuts down when he sees the topic is subtraction. I'm thinking we'll try 10 minutes of some sort of place value activity or easier subtraction, and then work on his regular curriculum, avoiding more complicated subtraction for a little bit.

[Not_a_Number]

2 minutes ago, knitgrl said:

It's nice to know other kids have an issue with this as well. My son can do easy subtraction, but having to borrow trips him up. He would say that 33-19=26, because 9-3=6 and 3-1=2. So I guess we need to work on place value some more. It's challenging because he almost completely shuts down when he sees the topic is subtraction. I'm thinking we'll try 10 minutes of some sort of place value activity or easier subtraction, and then work on his regular curriculum, avoiding more complicated subtraction for a little bit.

Give him manipulatives?

[knitgrl]

I think I probably need to give him a few extra days to forget about the horrors of subtracting double digits, and we'll try again with manipulatives. If that doesn't work, I'll be back. ;-)

[Not_a_Number]

1 hour ago, knitgrl said:

I think I probably need to give him a few extra days to forget about the horrors of subtracting double digits, and we'll try again with manipulatives. If that doesn't work, I'll be back. 😉

I always use poker chips as a manipulative nowadays 🙂 

[EKS]

I found that doing mental math with double and triple digits helped to solidify the idea of subtraction (and the other operations as well).

Start with something that doesn't require regrouping, say 76 - 31.  It's ok to write it down (horizontally) and let him look at it while he does the problem.  He should say "Seventy-six minus thirty is forty-six, and forty-six minus one is forty-five."

Once he's good at this, you can move onto problems that require regrouping, say 82 - 56.  He should say "Eighty-two minus fifty is thirty-two, and thirty-two minus six is twenty-six."  He can even break down that last piece further if necessary: "Thirty-two minus two is thirty and thirty minus four is twenty-six."

Then you can extend this to three digit numbers, first without regrouping, then with regrouping in one place, and once he's really good, regrouping in two places and/or across zeros. 

The reason you have him actually say aloud what he is doing is twofold.  First, you don't want him simply doing the subtraction algorithm in his head.  You want him to think about it in terms of place value explicitly, and saying it this way forces that.  Second, doing things orally is a way to reinforce proper sequencing. 

Edited August 16, 2021 by EKS

[Not_a_Number]

1 hour ago, EKS said:

The reason you have him actually say aloud what he is doing is twofold.  First, you don't want him simply doing the subtraction algorithm in his head.  You want him to think about it in terms of place value explicitly, and saying it this way forces that.  Second, doing things orally is a way to reinforce proper sequencing. 

I think this only works for kids with decent working memory, though. Some of the kids I teach would absolutely not be able to do this. 

[knitgrl]

2 hours ago, EKS said:

I found that doing mental math with double and triple digits helped to solidify the idea of subtraction (and the other operations as well).

Start with something that doesn't require regrouping, say 76 - 31.  It's ok to write it down (horizontally) and let him look at it while he does the problem.  He should say "Seventy-six minus thirty is forty-six, and forty-six minus one is forty-five."

Once he's good at this, you can move onto problems that require regrouping, say 82 - 56.  He should say "Eighty-two minus fifty is thirty-two, and thirty-two minus six is twenty-six."  He can even break down that last piece further if necessary: "Thirty-two minus two is thirty and thirty minus four is twenty-six."

We'll try this, too. In a few days. This is the child who refuses to cooperate for anything that is new or perceived of as difficult. After last week, he's decided math is hard. He enjoyed the ten minute activity this morning, and balked at the regular math curriculum, but slightly warmed up by the end because ordinal numbers and Roman numerals did not prove to be difficult. I'm hoping a few more days of math success and we can gently focus on subtraction.

[EKS]

3 hours ago, Not_a_Number said:

I think this only works for kids with decent working memory, though. Some of the kids I teach would absolutely not be able to do this. 

To help with working memory, I would (and did) write down intermediates.  You still get the benefit of the process but without the angst.  And I'm all about avoiding angst.

[Not_a_Number]

4 minutes ago, EKS said:

To help with working memory, I would (and did) write down intermediates.  You still get the benefit of the process but without the angst.  And I'm all about avoiding angst.

I found that manipulatives made kids less angry about the whole process, lol. But yes, I can see how writing it down would work, too! 

[EKS]

1 minute ago, Not_a_Number said:

I found that manipulatives made kids less angry about the whole process, lol. But yes, I can see how writing it down would work, too! 

We certainly used manipulatives, but it was before we got to this stage.  I used this sort of mental math in place of fact practice after fact practice became angst inducing.  Are you detecting a pattern here? 😆

[Not_a_Number]

51 minutes ago, EKS said:

We certainly used manipulatives, but it was before we got to this stage.  I used this sort of mental math in place of fact practice after fact practice became angst inducing.  Are you detecting a pattern here? 😆

I think I just let kids use the manipulatives until there's total fluency? So we spend a lot of time with them. 

[bookbard]

8 hours ago, EKS said:

Once he's good at this, you can move onto problems that require regrouping, say 82 - 56.  He should say "Eighty-two minus fifty is thirty-two, and thirty-two minus six is twenty-six."  He can even break down that last piece further if necessary: "Thirty-two minus two is thirty and thirty minus four is twenty-six."

I like this. I really do think one of the big issues has been all the different ways he's had to do the same problem, so when he gets something new, he doesn't know where to start.  

[Not_a_Number]

24 minutes ago, bookbard said:

I like this. I really do think one of the big issues has been all the different ways he's had to do the same problem, so when he gets something new, he doesn't know where to start.  

The way I deal with this is by having a single consistent mental model.

[Btervet]

I don't know if this will help, but it did help my kid who kept subtracting backwards, is I talked to him about how we "read" math left to right, top to bottom, just like we read words. This really clicked with him, especially because we had spent a significant amount of time teaching him to read left to right as he would try to sound out words backwards often. So when doing 33-19, I would just remind him that even in the ones place, you read it left to right as 3 - 9.

Another approach is to go ahead and teach negatives, and teach that 33 - 19 is really 33 + (-19). That worked for my other kid who would subtract incorrectly. So he would do the problem as 30 + (-10) to get 20, and 3 + (-9) to get (-6). Then 20 + (-6) = 14.

[kiwik]

On 8/13/2021 at 11:04 PM, purpleowl said:

You mentioned that he understands you end up in negative numbers - does he have an understanding of negative numbers? My son would look at 123-39 and say that's one in the hundreds, negative one in the tens, and negative six in the ones. Then he says okay, one hundred and negative one tens is ninety, so that's "ninety-negative six," and that's eighty-four. Wondering if that approach would work for your son as well? I know it's an odd way to do it, lol.

One of mine used to do that too.  

[kiwik]

Do you teach him to rewrite the problem.  123 - 39 = 124-40 = 100-40 +24 = 60+24 =84?

[Not_a_Number]

35 minutes ago, kiwik said:

Do you teach him to rewrite the problem.  123 - 39 = 124-40 = 100-40 +24 = 60+24 =84?

I find that kids need to be really ready to understand why this works. It's not one of the easier tricks. Like, thinking about increasing both the numbers by 1 requires a LOT of comfort. 

[kiwik]

41 minutes ago, Not_a_Number said:

I find that kids need to be really ready to understand why this works. It's not one of the easier tricks. Like, thinking about increasing both the numbers by 1 requires a LOT of comfort. 

You're probably right it is just a thing that was taught to my kids in school from pretty early (maybe 6 or 7?). Easier questions at first like 9+4=10+3. I don't think they ever really taught it any other way though I may be wrong.  They may do 123-40+1 rather than the way I did it but they were big on pulling numbers apart and moving bits around.  If course it was a complete disaster for a lot of kids so they are probably doing something else now.  The kids who got numbers did it anyway and the ones who didn't get it were completely lost.  The ones in the middle learned the technique.  Pretty much like any other method really.

[Not_a_Number]

7 minutes ago, kiwik said:

You're probably right it is just a thing that was taught to my kids in school from pretty early (maybe 6 or 7?). Easier questions at first like 9+4=10+3. I don't think they ever really taught it any other way though I may be wrong.  They may do 123-40+1 rather than the way I did it but they were big on pulling numbers apart and moving bits around.  If course it was a complete disaster for a lot of kids so they are probably doing something else now.  The kids who got numbers did it anyway and the ones who didn't get it were completely lost.  The ones in the middle learned the technique.  Pretty much like any other method really.

My kids can generally move numbers around, but some things are easier than others. I think the first thing I can generally teach kids is to move numbers from one number to the other in addition. With subtraction, it's fairly easy to subtract one piece and then another, but not so easy to subtract too much then add on. And adding 1 to both is quite a pain to conceptualize. 

Anyway, I am pretty sure if a kid was switching around the digits however they felt like it, they wouldn't really understand this trick, since their understanding of subtraction is definitely very iffy. 

[knitgrl]

On 8/16/2021 at 4:24 PM, knitgrl said:

We'll try this, too. In a few days. This is the child who refuses to cooperate for anything that is new or perceived of as difficult. After last week, he's decided math is hard. He enjoyed the ten minute activity this morning, and balked at the regular math curriculum, but slightly warmed up by the end because ordinal numbers and Roman numerals did not prove to be difficult. I'm hoping a few more days of math success and we can gently focus on subtraction.

What we have done so far....

I pulled some MM practice sheets from 2nd grade and let him have success with easy things, and are working up to harder things. I tried using the RS abacus, which sort of helped, but not really. Poker chips did not work. We have some 18x24 inch white boards, and I counted out beans onto the white board and circled groups of ten to demonstrate problems. That sparked something. However, I did not want to count out 80 beans, so I used base ten blocks and that seems to be clicking, but I think we will be practicing a bit with that before we moved forward.

However, the main thing that has turned this around is SWB's recommendation of bribing with chocolate chips. For each page, he gets 6 chocolate chips for having a good attitude, and 10 if he gets them all right which has been good for reducing careless errors. Subtraction has gone from being something he completely dreads to something that has at least one element of pleasure to it.

[Not_a_Number]

11 minutes ago, knitgrl said:

Poker chips did not work.

What happened with poker chips? 

[bookbard]

Thanks to whoever revived this - I was going to say the subtraction thing is fine now, he can do 3 digit with regrouping, no prob. Guess he was just at a developmental stage or something. 

[knitgrl]

16 hours ago, Not_a_Number said:

What happened with poker chips? 

He was not engaged, but I confess I think I tried this before using chocolate chips, so that might have something to do with it.

This child spent 6 months on CVC words. Once it clicked, his reading took off and he did not need that much more phonics instruction. So I am hoping this is somewhat similar, and once he really gets it, we can move on. Also, his older sister, who also had issues with regrouping, has taught him weird ways of subtracting, like if you want to subtract 9, you take away three 3's. I have no idea how to undo that.

Edited September 24, 2021 by knitgrl

[Not_a_Number]

14 minutes ago, knitgrl said:

He was not engaged, but I confess I think I tried this before using chocolate chips, so that might have something to do with it.

Sorry to bug you -- just curious what happened precisely! How were you using them? 

[knitgrl]

6 minutes ago, Not_a_Number said:

Sorry to bug you -- just curious what happened precisely! How were you using them? 

So if we were trying to work on 35-19, we had chips representing 10's and units, so 3 larger chips, 5 smaller chips. You can take away one ten, but since there weren't 9 units to take away, you had to swap one of the tens for ten units so that there are enough. He didn't want to hear it. I dunno. He can be very obstinate at times.

[Not_a_Number]

On 9/24/2021 at 2:52 PM, knitgrl said:

So if we were trying to work on 35-19, we had chips representing 10's and units, so 3 larger chips, 5 smaller chips. You can take away one ten, but since there weren't 9 units to take away, you had to swap one of the tens for ten units so that there are enough. He didn't want to hear it. I dunno. He can be very obstinate at times.

Ah. Got it. That probably works best before the kid has a lot of practice doing it in a faster way, I think. I generally have trouble backtracking, because kids get so impatient.