Loving your child's mind: Mental math for fun

Hello, Michael Strong.

Good morning, Alana, how are you?

I'm good, how are you?

I'm well, I'm well. Um, I thought I'd check in on math these days. What are you learning in mathematics these days?

Um, right now I have the exact same workbook from school that I brought. Um, it's mostly— I'm mostly practicing like adding 3 3-digit numbers, subtracting numbers, stuff like that.

All right, how's it going?

It's going pretty good.

All right, so 3-digit numbers, that's— I, I remember when we were doing just, you know, simple, simple numbers. Um, so would you say that adding 3-digit numbers is pretty similar to adding single digit and 2-digit, or does it get different when you get to 3 digits?

For 3 digits, to me it's kind of a little different because when you're borrowing, it's kind of different. You have to borrow from the— sometimes if you have, like, you have to borrow from the tens and the ones too. It makes, like, for me sometimes I have to start at the hundreds, not the tens or anything like that.

Have you ever done any 4-digit numbers or 5-digit numbers?

No.

Do you think you could? Do you think you could do a 4-digit?

I don't— I'm not very good with that, so probably not.

Okay, um, so would you say, uh, what's the lowest 4-digit number you can think of? Do you know what the lowest 4-digit number is?

Um, maybe 1,000.

Yep, 999. It's the last 3-digit number, and 999 plus 1 is 1,000, so that's the first 4-digit number. So what is 1,000 plus 1,000?

1,000 plus 1,000 is 2,000.

You just added two 4-digit numbers, didn't you?

I know, that was easy because it's kind of just like 1 plus 1 or, or 20 plus 20 or 10 plus 10, except it's just in the thousands.

Fair enough, fair enough. How about, um, 1,500 plus 1,000?

2,500?

You just added another 4-digit, set of 4-digit numbers, right?

I guess.

So was that one easy as well?

Yeah, I think it was pretty easy because you all, because you, because if it was in the hundreds, you, you just have to add, um, 150 plus 200. Which is just 250.

How about 1,500 plus 1,500?

Can you say that again? You cut off for a little bit.

Yeah, I know, the screen froze. 1,500 plus 1,500.

Oh, 1,500 plus 1,500 is, um, 2,020.

How do you get that? Explain how you came up with that.

Okay, wait, what was the equation again?

1,500 plus 1,500, or said differently, 1,500 plus 1,500.

Wait, no, that'd be 3,000. Right?

That's right, that's right. How did you do that one?

Because when you just said like 1,500 plus 1,500, I know that the thousands equal to 2,000 and the 500 plus 500 is another 1,000, so you just add 2— sorry, 2 plus 1, which is 3,000.

That's exactly right, that's exactly right. So it sounds like there are a bunch of 4-digit numbers you can do. Even in your head, right?

Yeah.

So how about 1,200 plus 1,800?

1,200 plus 1,800 is 1,100.

So what's the total?

1,100. Sorry. Yeah, 1,000.

So tell me how you figured that out. What did you do?

Wait, can you tell me the equation again?

I forgot again. So 1,200 plus 1,800.

Oh, I thought you said 1,800. Um, okay, so 1,200 plus— wait, what was the other one? 1,800? Um, I know that 2 8 is 10, so, and 1,000 1,000 is 2,000, so I think that would be 2,100.

2,100?

Or 3,000.

Um, it took you a minute to kind of figure out that the 1,000 is 1,000, right? Is that, is that what happened? It's interesting how we hold numbers in different ways in our heads because, you know, and even the way I'm asking the problems. So, you know, the numeral 1-5-0-0 can either be 1,500 or 1,500. You can say it both ways, right?

Yeah.

And so I think some of this is translating how we're saying it into— I don't know, do you figure it out using words or do you put it into numerals or how do you, how do you kind of process it in your brain?

Um, I really just break it down, down to easier equations. So if you were saying like 4,000 plus 4,000, I know that 4 plus 4 is 8, so that'd be 8,000.

Mm-hmm. Yep. That's a good way to do it. Um, in which case, how about, uh, Let's do 2,200 plus 2,200.

2,200 plus 2,200. I'm guessing 800 and 2— 8,200.

And how did you get that?

Um, I just added what my brain was thinking is I just added it all up. So like the 220 would be 4, and then you said, um, like 2 plus 2 again, which would be— wait, no, 600 and some— sorry, 6,000 and something, right? I don't know.

So you went from 8,000 to 6,000. Let's— so let's go more slowly. One of the things is, in math problems, do you ever go back and check your work to make sure it's correct?

Yeah, like, for example, when I was in school, when we were done with our workbooks, we'd go— if you had a 4th grade notebook, you'd have to go to my teacher, Emma, so she could check it. And if you had a 3rd grade notebook, you had to go check it with this piece of paper that held that had all the answers on it.

Mm-hmm. But do you ever check your own work by recalculating to make sure you calculated correctly?

Um, no, we don't.

Okay. Do you see why that might be useful sometimes?

Hmm, I don't really think so. Or how would it be useful?

Okay, so that's a good question. What I'm thinking is that, um, what do you think causes mistakes in mathematics? Why do we— so do you get every problem right all the time, or occasionally do you get some problems wrong?

Um, maybe, um, kind of rarely I get a question wrong. Like sometimes, like, if I, like, I had a really, really hard question that I really didn't know anything about, of like like maybe on ST Math, and I would try to make my best guess, and sometimes I get those wrong.

I see. And you said ST Math, is that what you said?

Yeah, sometimes I go into ST Math to practice my math too because since I'm going on to online school now.

I see. Um, and so when you say it's, uh, sometimes you get it wrong and you're guessing, when you, uh, try to figure out We started with 1,000 plus 1,000, right?

Yeah.

Would you say you guessed that it was 2,000 or that you knew that it was 2,000?

I knew that it was 2,000 because I know that, for example, 10 plus 10 is 20. So if you just add that, that'd be 2,000.

So what's the difference between knowing the answer and guessing the answer?

Maybe guessing, like for example, at the beginning of this year when we were doing tests on our Chromebooks, it asked me a few division questions and I seriously didn't know the answer. So I just took my best guess.

Okay. And so when you're doing right now, you said you're learning 3-digit addition and subtraction. Do you have to guess when you're doing those problems or can you figure it out?

I can figure it out if I like, if it's typed out or written down or printed out and I have some time to do it like Like maybe 20 seconds, I don't know. I could probably just do that problem because most of the time if I need a strategy I just use regrouping.

And then even with figuring it out, so on those kinds of problems you know how to figure it out, do you occasionally get an answer wrong or do you always get the answer absolutely right every single time?

Um, I don't necessarily sometimes get answers wrong on that. Like, I usually get answers wrong if, like, it was a, like, maybe if, like, a triangle equals, like, 10 or something equals, like, 20. And then if a square plus triangle equals what? Sometimes I don't get that. So I get that one wrong.

I see, but with 3-digit addition you always get those correct, is that right?

Most of the time, yes.

Okay, so let's go back to 2200 plus 2200, which let's say it as 2200 plus 2200. Um, what— how should we solve this? How do we figure it out?

Uh, oh, um, So you have to add the 2 thousands together, which is 4,000, and then the 2 20 hundreds. Am I correct?

2 hundreds. So 2,200 plus 2,200.

Okay, so that'd be 400. So it'd be 4,400.

Exactly right. So initially you came up with an $8,000 guess and then a $6,000— not dollar— an $8,000 guess and then a $6,000 guess. And now the right answer is 4,400. So why do you think you came up with, um, 8,600— I mean 8,000 and 6,000 before we got to the 4,400?

Um, I'm guessing because maybe I just thought about it wrong.

Okay, it seems like when I say it as two thousand two hundred, two thousand two hundred, that's easier than when I say twenty-two hundred. Is it— does it seem accurate to you that it's more confusing for you when I say 2,200? So let's, let's do one more and I'll—

Yeah, it is because—

Go ahead, because why?

Okay, um, I know that like, I know that 10,000 is a thou— sorry, 1,000 is 1,000 and that's 2 10-hundreds, sorry, yeah, 2 10-hundreds plus another 2 hundreds. And for me, before that was kind of hard to process.

Yep. So how about let's try to add 5,100 plus 5,400.

5,000— is the answer in the millions?

No. So 5,100 plus 5,400.

So it's 500, 200, 5,200 plus 5,000 plus 5,400?

Uh, 5,100 plus 5,400.

Okay, that'd be 10,500.

Exactly right. So why did you think it was going to be in the millions first?

Because I didn't know that, like, um, if you had 10,000, I thought that if you meant another thou— if you want another thousand, then that'd be million. I'm not good with my millions.

Okay, actually it goes, um, 1,000, 10,000, 100,000, and then a million. So, uh, a million is you have to go all the way through the 100,000s first. Okay. And then would you say you hold numbers in your head when you try to do these calculations?

What do you mean by that?

So it sort of sounds like, um, we'll go to the 1,500 plus 1,500. Sounds like you hold 1,000 and 1,000 in your head over here and makes 2,000. Then you hold 500 and 500 in your head and it makes 1,000. You take the 2,000 and the 1,000 and put them together and you get 3,000. Is that kind of what you're doing in your head?

Yeah, like for example, the problem that you just gave me, I knew that 400 plus 100 is 500 and I knew that 5 plus 5 is 10, so it'd be 10,500.

Yeah, and I mean holding it in your head 'cause sometimes I see you looking off like this when you're thinking. And somehow I think you're kind of holding numbers in your head. Does it seem strange to describe it as holding numbers in your head, or do you have a better way to describe it?

Um, not really, but I do know that I used to just like look up, try to look up in my brain.

No, it's very funny. Um, that's why I, I can't think of a better way to say it than hold numbers in my head. And I would say when I make mistakes, it's because I'm trying to hold numbers in my head and then I can't hold them all and like they slip away. So, you know, if you get too many— and there are people that can hold, you know, lots and lots and lots of numbers in their head. And I think, you know, there's a whole— what we're doing is kind of mental calculation. That is, you can do calculations on pencil and paper, but mental calculation is when you do it all in your head. And there are people that get really, really good at this, and they can calculate, say, 3,675,243 times 8,643,745, something like that, instantaneously.

I couldn't do that. I can't do that.

Of course, and right now I can't either. But what I think is a lot of it is practice. That if you practice at anything you get better at it? And so even with the, um, you know, when we started you said you knew how to do 3-digit numbers, you did not know how to add 4-digit numbers, is that correct?

Yeah.

And so then we did a bunch of 4-digit numbers, so clearly you can add some 4-digit numbers in your head, right?

Some, yeah.

So do you think that with practice you could get, and you say you, you pretty much always get the 3-digit numbers correct adding 2 3-digit numbers. Is that only on paper or could you also do it in your head?

I could also probably do it in my head. It would be a little more easy if it was addition. I'm, I'm from, I think that I'm a little better with my addition than subtraction.

Yeah, and, and I think part of it is I think In subtraction, you have to hold more in your head, basically. So let's just try a 3-digit number. How about 321 plus 243?

Can you say that again?

321 What's 500 plus 243?

564.

And how did you do that?

It's because I added the hundreds, which would be 500, and then the tens, which would be— wait, whatever I said— and then the ones, which would be 4.

Okay, and it was 321 and 243, and so the tens were 2 and 4, which was 6. And so that's why in holding in your head right there, you forgot the tens. It sort of— that's what I mean by without practice it slips out of our head, but I think with practice you can hold lots of numbers in your head. Well, cool, it's, it's fun to— would you like doing math?

I, I really like doing math now.

Excellent. No, I think it is fun. And one of the reasons I think it's useful to check your own work is once you know the rules— so, you know, with complex division you don't know the rules, so of course you have to guess. But with adding 3-digit numbers, you do know the rules, and so you could check your own work just to make sure you didn't make mistakes. And I think especially with mental math— you know, with paper math you can kind of check to make sure, did I do everything correctly. But with mental math, it's easier for the numbers to slip out, and so you have to develop ways to make sure you can check it to make sure it makes sense at the end. So maybe we should probably stop for today, but maybe in the future at some point we'll play around with mental math a little bit more. Does that sound good?

Yeah.

All right, all right, have a good day, Alana. Take care. Bye-bye.