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DS wants to know more about why we do all the steps in long division. He understands the general concept if we have to divide 96 by 4 = 26 (a group of 96 into 4 equal groups equals 26 in each group). But he wants an intermediary explanation step between that and why we do all the steps in long division.

 

I did see a website by Ashleigh-education journey.com that is promising. But I'd like to research a bit more to discuss with him. Any resources are appreciated. Manipulative explanations might help better too.

 

Edited because math is hard 🤣

Edited by displace
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96 divided by 4 is 24.

 

It's all about place value. It is easier to explain if you have place value blocks.

If you have 96 in place value blocks, you have nine rods (10s) and six cubes (1s).

 

Start sorting the largest place value into four equal piles.

You can put two rods into each of the four piles, so you write a 2 over the 10s place value in long division.

Subtract out those eight rods from the original nine rods. You have one rod left over. This is the subtraction in long division.

 

Convert the rod to cubes and combine it with the original cubes.

You now have 16 cubes: the original 6 and the ones from the rod.

This is like bringing down the next place value.

You can put four cubes into each of the four piles, so you write a 4 over the 1s place value in long division.

Subtract out those sixteen cubes that you just put into the piles from what you had.

 

You have no place value blocks left, so everything divided neatly with no remainder.

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I like this Cookie Factory Explanation.

 

For a while, I was working one problem that way on the white board every morning while all the kids watched and helped.  Even the 5 year old started to get the hang of it.

 

Eventually, we got bored of cookies, and started trying to divide other things using the same procedure.  ie. There were 9,876 houses that needed to be painted, and 7 painters were going to divide them up evenly.  Each street contained 10 houses, each neighborhood contained 10 streets (100 houses), and each city contained 10 neighborhoods (1000 houses)...etc.

 

Wendy

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96 divided by 4 is 24.

 

It's all about place value. It is easier to explain if you have place value blocks.

If you have 96 in place value blocks, you have nine rods (10s) and six cubes (1s).

 

Start sorting the largest place value into four equal piles.

You can put two rods into each of the four piles, so you write a 2 over the 10s place value in long division.

Subtract out those eight rods from the original nine rods. You have one rod left over. This is the subtraction in long division.

 

Convert the rod to cubes and combine it with the original cubes.

You now have 16 cubes: the original 6 and the ones from the rod.

This is like bringing down the next place value.

You can put four cubes into each of the four piles, so you write a 4 over the 1s place value in long division.

Subtract out those sixteen cubes that you just put into the piles from what you had.

 

You have no place value blocks left, so everything divided neatly with no remainder.

This may help. I was using individual manipulatives instead of place value blocks.

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I like this Cookie Factory Explanation.

 

For a while, I was working one problem that way on the white board every morning while all the kids watched and helped. Even the 5 year old started to get the hang of it.

 

Eventually, we got bored of cookies, and started trying to divide other things using the same procedure. ie. There were 9,876 houses that needed to be painted, and 7 painters were going to divide them up evenly. Each street contained 10 houses, each neighborhood contained 10 streets (100 houses), and each city contained 10 neighborhoods (1000 houses)...etc.

 

Wendy

Thanks. Those are good explanations. We do 1-2 problems per day for practice.

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DS wants to know more about why we do all the steps in long division. He understands the general concept if we have to divide 96 by 4 = 26 (a group of 96 into 4 equal groups equals 26 in each group). But he wants an intermediary explanation step between that and why we do all the steps in long division.

 

I did see a website by Ashleigh-education journey.com that is promising. But I'd like to research a bit more to discuss with him. Any resources are appreciated. Manipulative explanations might help better too.

 

Edited because math is hard 🤣

You need to use the 1s 10s and 100sblocks if you have them. Show him dividing the 100s. Exchange the leftover 100s for 10s. Divide the 10s. Show exchanging the leftover 10s for 1s. Do the steps physically as you do the process on paper.

 

Also it can be tricky I find with problems like this because we tend to teach using easy problems so they can learn the process but then the smart kids look and think - well why do I need all that when I can just do it on my head? Sometimes you need to throw a harder problem at them so they see the need to follow the process.

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Another great tool is the Montessori long division beads and board. It is somewhat spendy but I have seen it cheaper on discount Montessori suppliers. It is hands down the absolute best tool I have seen to help kids concretely grasp long division.

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Another great tool is the Montessori long division beads and board. It is somewhat spendy but I have seen it cheaper on discount Montessori suppliers. It is hands down the absolute best tool I have seen to help kids concretely grasp long division.

I've seen those before but I'll have to look into it further. I assume there are videos explaining how to use them. And I'm trying to downsize... (which I say all the time). 😆

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Oh I really like that cookie explanation link too. I did not know about that one. I like to start with the manipulatives and I like showing it as a rectangle as in the Education Unboxed video and also by distributing the place value blocks and Cuisenaire rods to the number of people you are diving by.

Edited by MistyMountain
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Coming back to say, that I think one of the most important tools for really understanding long division is being asked to figure out, sans algorithm, how to divide a large number into a certain number of groups.

 

No matter how you approach it, the process is always fundamentally the same: put a certain number of objects into each group and then figure out how many are left.  Repeat until you run out of objects...or, at least, until you have less objects than groups.

 

DS and I actually play acted this out, to really experience the steps.  We pretended like DS was the lottery, and me and the other kids were all winners of a big jackpot.  The lottery now had to split the total winnings among the 4 of us.  We made up a really big number for the total jackpot, and then DS pretended to hand each of us money while telling up how much we were getting.  Then he would make notes on his ledger sheet to figure out how much more he still had to divide.

 

We played with similar scenarios until DS was very comfortable with the steps...not on a memorization level, but on a conceptual level.

 

Wendy

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Coming back to say, that I think one of the most important tools for really understanding long division is being asked to figure out, sans algorithm, how to divide a large number into a certain number of groups.

 

No matter how you approach it, the process is always fundamentally the same: put a certain number of objects into each group and then figure out how many are left. Repeat until you run out of objects...or, at least, until you have less objects than groups.

 

DS and I actually play acted this out, to really experience the steps. We pretended like DS was the lottery, and me and the other kids were all winners of a big jackpot. The lottery now had to split the total winnings among the 4 of us. We made up a really big number for the total jackpot, and then DS pretended to hand each of us money while telling up how much we were getting. Then he would make notes on his ledger sheet to figure out how much more he still had to divide.

 

We played with similar scenarios until DS was very comfortable with the steps...not on a memorization level, but on a conceptual level.

 

Wendy

I've done the same thing, but with a pretend bank robbery.

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Instead of buying the 100's, 10's, and 1's blocks, you can make them. I highly recommend using them. 

Years ago I did a long division boot-camp at a private school, every morning for two weeks  the students who were just stumped came to me. Every kid had their own set of blocks (made from paper), a white board and marker and eraser. We did 3 problems together - each step, and then three problems by themselves. All but one kid had it mastered the first week, the second kid needed the extra week, but he got it. 

 

By the way, I also did the same thing for kids who were struggling with borrowing. (I usually had the same kids in both groups a few years apart.)

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I don't even know if you need c-rods or base ten blocks to explain why the steps in long division work. Assuming, of course, he understands the concept of division in general. C-rods will definitely help with that.

 

If we use something easy like 999 / 3, you're basically just doing (900 / 3) + (90 / 3) + (9 / 3). 300 + 30 + 3 = 333. You're dividing the highest place value first, check to see if there's a remainder, then move on to the next place value, check to see if there's a remainder, and repeat until done.

 

If he needs help grasping the concept, have him expand the dividend, divide each number separately, and then add the quotients. He'll get the same answer as if he does it with long division because long division is a more compact way of doing the exact same thing.

Edited by Mergath
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Every kid had their own set of blocks (made from paper), a white board and marker and eraser.  

 

 

 

OMG, I totally did this with my kids, since we were using Singapore Math at the time.  I had little squares of construction paper labeled with "1" "10" and "100."  We could slide them around on this large white board on the floor, which had 3 columns for 100s, 10s, and 1s.  And we did as Mergath described above, dividing up the 100s and finding remainders, replacing the remainder 100s with 10s and adding them to the 10s column and so on.  

 

At the same time I had the long division notation going on the side, we could see what was happening to my pieces of construction paper as we were writing out the numbers.  

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Instead of buying the 100's, 10's, and 1's blocks, you can make them. I highly recommend using them. 

Years ago I did a long division boot-camp at a private school, every morning for two weeks  the students who were just stumped came to me. Every kid had their own set of blocks (made from paper), a white board and marker and eraser. We did 3 problems together - each step, and then three problems by themselves. All but one kid had it mastered the first week, the second kid needed the extra week, but he got it. 

 

By the way, I also did the same thing for kids who were struggling with borrowing. (I usually had the same kids in both groups a few years apart.)

 

I think that 100's, 10's and 1's blocks are hugely useful in the early stages of math, but in my experience most kids who are advanced enough to do long division have internalized the idea that 10 equals 10 ones, and the idea of trading and exchanging, well enough that they can also use play money if the blocks are missing.  

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I don't even know if you need c-rods or base ten blocks to explain why the steps in long division work. Assuming, of course, he understands the concept of division in general. C-rods will definitely help with that.

 

If we use something easy like 999 / 3, you're basically just doing (900 / 3) + (90 / 3) + (9 / 3). 300 + 30 + 3 = 333. You're dividing the highest place value first, check to see if there's a remainder, then move on to the next place value, check to see if there's a remainder, and repeat until done.

 

If he needs help grasping the concept, have him expand the dividend, divide each number separately, and then add the quotients. He'll get the same answer as if he does it with long division because long division is a more compact way of doing the exact same thing.

 

The thing is that most problems end up with those remainders and the algorithm obscures what's happening for a lot of the kids. So they have 729 / 3 instead and in the first step - 700 / 3 - it's easy to see that the 3 goes 200 times... except, it goes a lot more than that and you've got 100 left over, but the way you write it, it looks like a "1" that you're regrouping and then you "bring down" the next number, which is the 2, so now you're dividing the 3 into 120, but why didn't you just do it into that 100 because it would have fit more times and... I think it's easy for kids to lose track of all this. I think it's easier than when they're adding or subtracting (even with regrouping) or even multiplying. There's something about the steps in long division that makes it harder to connect the meaning to the algorithm. I think it's one of the first more complicated algorithms that they can't just immediately reconnect to the why of by looking at it that kids hit on. And it becomes even more complicated when you have two and three digit numbers in the divisor. So I think base 10 blocks or C-rods help kids see that.

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