Learn about Singapore Math: Grade 3a Unit 2 - part 2
Let’s keep doing some of the Singapore math problems and I’m on unit two and this is part two, not officially part two but I just did unit two part one so from my point of view it’s part two but the problem we are going to do is problem four that is on page 20 and it says, Mary made 686 cookies. She sold some of them. If 298 were left over how many cookies did she sell? The hardest part about this is writing it down which you don’t have to do because it’s already on your book. How many cookies did she sell? Maybe you think that the hardest part about this is reading my hand writing, perhaps.
Let’s see if we can draw this out. Well, she made 686 cookies so let’s draw like a bar that represents all of the cookies that she made. So let’s say that this bar has a length 686 where the units are cookies instead of centimeters. Well, maybe that should be an international standard of length of the cookie. But anyway let’s say that this bar has a length of 686 and it represents the total number of cookies she made and then she said sold some of them. We don’t know exactly how many she sold. So the number that she sold let me just shade that in here. This is how many she sold and we know that after she sold a bunch she had 298 were left.
So 298, if I’m just to approximate that that’s about 300 it a little bit less than ½ so, roughly this much was left so that’s the 298. So this length is 298. So the question is asking us how many cookies did she sell? Well the 298 is how many she had left over and so what she sold must be this blue chunk. This is what she sold. I hopefully makes a little of intuitive sense for you. If you make 686 and you’re left with 298, the amount that you sell must be the difference between 686 and 298 and so what is the difference for this blue region between 686 and 298.
Well, we learned in the previous video that that is just equal to 686–298. This whole thing minus this is equal to the amount that she sold and you could do that math and actually what I’m going to do here and I’m assuming that this is covered in Singapore math grades, probably grade two but I’m actually going to review it just because I think it’s interesting and frankly I think once you know how to do subtraction and regrouping with three digits this three must be the same thing to do with four or five or six or seven which will do shortly but I just want to make sure you really understand the intuition here.
So, if we can figure out what is 686–298 is we have solved the problem. We have figured out exactly how many cookies Mary sold. So let’s see if we can do that. So let’s figure out what’s 686, so it’s 686–298 is and I’m going to go back to what we did with all of the places and our buckets of barrel of marbles and all of that because I really want to give you the intuition of what’s happening when we subtract these two numbers so what’s happening when we do things like borrowing or regrouping.
So, both of these are three digits numbers. We have our ones place or tens place and a hundreds place, so let’s draw those columns and I forgot the colors that I used previously so let’s just say the hundreds place. This is hundreds, this is tens and this is ones.
So we’re starting off with 686 in the case of this problem, it’s 686 cookies so what does this six represents? This essentially represents six ones. So we could draw them here one, two, three, four, five, six and these are all ones. What does this eight represent? Well, those are tens, so we have eight tens, one, two, three, four, five, six, seven, eight and each of these represent ten cookies or ten things or whatever we’re measuring. These are tens.
And what does this six represents? Well it’s six hundreds, so if we look at a hundreds place we could view this maybe on the top of cans, one, two, three, four, five, six and each of these represent a hundred, each of these represent one hundred and we’re going to take away 298, 686–298, so how do we that? Well our first temptation might be will let’s just take away eight ones but we don’t have eight ones here we only have six, so how can we take away eight ones?
Well, what we could do is we could take one of these tens and turn it into ten ones, so what we could do is let’s take this ten and convert it to ten ones. All we’re doing is we’re opening up this can of marbles maybe when you're using cookies in this example and instead of having a can of ten we’re going to take out all the marbles and put them as ones. So let me do that. So I’m going to cross this out here and I’m going to put ten marbles here or ten cookies and all these are ones.
Where did I get them from? Well, from whatever, I am just drawing this dotted line to show that I got them by exchanging this right here. So how many ones do I have now? I’ve ten plus six so I’ve 16 ones now. So now I can take away eight of these. So I had 686 minus 298.
So what did I just do? I took away one of this eight tens so now I only have seven left and I convert it to ten ones, so now I have 16 ones. So those of you familiar with borrowing I borrowed one from the eight or we could we say we regroup. We took ten and we put it into the ones group. In either way hopefully all of this will make sense to you because there were all just different ways of doing the same thing.
Now we can take eight away. We have 16 ones. We can take eight away so what happens when we take eight away? And then how many do we have left? We have eight left so we have eight left. That’s my tens. I want to free up some space so I could do this relatively neatly. Let me make that line so this is tens so I have some space, so that is tens. Okay, so what did we do? We borrowed ten from the tens place so we could say we cash in this ten bucket and we got ten ones here and that’s where the six became 16 and then since we used one of this tens we now have seven left in the tens bucket, fair enough.
We have 16–8 = 8. We have 16. We got rid of eight and we have eight but now we have seven tens, and that’s represented by seven here. But we want to subtract out nine of them, so once again we don’t have enough tens. So what could we do? Well, we could do the same. I want to call it a trick because there’s nothing tricky about it. We could take in and we could cash in one of this and get ten tens for them so let me do that.
So let’s take this one and convert it to ten tens, so we have ten and which of these are ten? Each of those are ten and we cash that out so now what is our equation? What does our problem look alike? Well, how many tens do we have now? Well, how many hundreds do we have? We have five. We used to have six, now we have five and that makes sense because we took one of them. We took one of them and we convert it into tens.
And so how many tens do we have now? We have seventeen, so we have seventeen. Let me just put a one here. Actually I’d cross out the seven but we could just put a one here, so we have seventeen tens, right? So now if you’re familiar with the term borrowing that’s essentially what we did but we did really borrow. It’s not like we’re not going to give it back. That’s borrowing, we took it. We took it from the hundreds place. So we took these hundred canisters of cookies and convert it to ten canisters of ten cookies each. That’s what we did and so that’s why our six canisters of a hundred turn into five canisters but that’s why we got ten extra cookies in the tens place. Hope I’m not confusing you.
But anyway, now we can subtract. We have to take out nine of the tens. This is the tens place. So we have seventeen, let’s take out nine, so and so how many do we have left? We have eight left. We have eight left so that means seventeen minus nine is eight, fair enough. Now we go to the hundreds place. We have five hundreds left. We had six but we have to convert one of them into tens. So we have five of those but then we have to take out two hundreds, so let’s do that and how many do we have left? We have three so we have three hundreds left.
So 686–298 = 388 and you could see that here, 388. So hopefully that wasn’t confusing to you. I mean you might have already known how to borrow or a regroup like this but it just hopefully given you a little bit intuition.
But just remember all, everything I did just not to solve that original problem where Mary had made 686 cookies. She sold some of them and she had 298 left. So essentially we now know that she must have sold 388 cookies and you could view it the other way. You could say she start with 686 cookies. She sold 388 cookies and now she has 298 left and you could do the subtraction and confirm that it’s true and this backs up that the difference between 686 and 388 is 298 and the difference 686 and 298 is 388. Anyway I’ll see you in the next video.