Taking a look at the Long Division Algorithm

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Today I’m going to try to tackle the process of long division. This is a complex algorithm to learn and often is a sticking point for many children and their parents. It is usually introduced in fourth grade, but I waited until fifth grade with both of my to introduce it, because of its level of complexity. 

Review

To begin any new concept, but particularly with long division, we always want to look at what is involved for us to understand the concept. This tells us what we need to review before we start to increase the likelihood of understanding. For long division, there are several concepts that need to be understood, and it is valuable if you take some time to review those concepts before you start teaching long division. These are multiples, the idea of composition and decomposition of numbers, place value, and multiplying by magnitudes of ten.

Multiples

The first of those concepts is multiples. We learn multiples early on through skip counting and if we’ve done any work with finding the least common multiple, we’ve also looked at multiples to find the common multiples. Multiples come into play with long division, because we are looking for the closest multiple of a number to our answer particularly when we are looking at a place value.

Decomposition and Composition of Numbers

Another concept that is imperative in understanding long division is the idea of composition and decomposition of numbers. They must understand that we can pull apart and put together numbers to make calculations easier for us. In particular, we use expanded form, which, if you don’t remember, is when you break a number up by place value, telling how many hundreds, how many tens, and how many ones we have. For instance, if we were to expand 365, we would have 3 hundreds, 6 tens, and 5 ones or 300+60+5. It can also be written 3 x 100 + 6 x 10 + 5 x 1.

Place Value

Very closely related and intertwined with number decomposition and composition is place value. Place value is also a concept that is imperative to understand when approaching long division, because in the end, when we start looking at more complex problems, we are decomposing and composing numbers by place value. That is, we are usually separating and putting our numbers back together by place value. We do this with both the total and the quotient (answer or the whole number of times one can divide one number by another).

Magnitudes of 10

The last concept, very closely related to place value, is an understanding of multiplying by the magnitude of 10. This is simply the understanding when we multiply by magnitudes of tens, we are adding zero for each place value. So, any number multiplied by one gives you that number.  If you multiply by 10, you’re adding a zero at the end of those digits. When we multiply by 100, we’re adding two zeros at the end of those two digits. For instance, 23 x 1 gives is 23;  23×10 gives us 230; 23×100 gives us 2,300.

Inverse Relationships

This last idea is not really a concept, but it is also important that a child understand that division is the opposite of multiplication. That becomes a little bit clearer when we look at the area-model multiplication and division. If you’ve already explored multiplication through area model, it might be worthwhile to review that as well.

Starting Simple

When we have those concepts well practiced and understood, it makes the process of learning long division much easier. Like all concepts, I suggest, even if you’ve gone over division before that, you’re back up the little, and do easier problems with your child. These are problems that your child knows. In fact, because the question of division is so misunderstood, it’s helpful to go back to the very beginning — simple things like 12÷4.

The reason that I suggest doing this is because the question that we have for division is “How many fit within a total?”  For our earlier example, how many 4’s fit into 12? There are many ways to look at this and many manipulatives that can be used, but of course, my favorite is the Cuisenaire rods. These just depict the mathematical concepts so well. 

After exploring what division means in this light as there are different ways of looking at division, we start small and scaffold our way to more complex problems. From here we may want to look at two digits, divided by one digit in which it divides evenly in each place, and there aren’t any leftovers to move to the next place. For instance, 36÷3 or 48÷4. 

Area Model Division

With the rods, we have traditionally done this by building 48 into a long train by placing rods end to end and then placing the 4’s (purple rods) end-to-end underneath it to see how many 4’s fit into 48. We want to shift here to look at an area, model of multiplication and division. To do this, we can build the 48, and then use the purple four rods to count multiples to 48.  Arranging the 4’s (purple rods) into 48 and looking at a way that we can make a rectangle with four on one side and our quotients (answer) on the other. For the example of 48, we have 4 tens and 8 ones. We can clearly make a grouping of 4 tens with 4 on one side and 10 on the other. Then group the remaining 8 into two 4’s. This is our beginning of splitting up things into place values.

Using Place Value

From here we can begin to explore numbers that do not divide evenly into each place. For example, 56÷4. Here we have 5 tens and 6 ones we can take and make one grouping from the tens of fours giving us 4×10, but we have 1 ten that does not fit into that grouping. What happens to that ten? We move that tens over to our ones changing our 6 to 16. This is how many we have left we have used 40 and that leaves us with 16 left. This is where we see a clear shift to using place value.

Using Magnitudes of Ten

At this point, we look at larger numbers in the hundreds divided by a single number. In the video, I talk about exploring the problem and seeing what we know. For instance, if we have something like 582 ÷ 3, we can talk about what we know. We know 3×10 is 30 so it’s going to be more than 10 we know 3×100 is 300, so it’s going to be over 100. However, if we double the 300 by multiplying the 3 by 200, we have 600 which exceeds our total of 582, so we know it’s going to be less than 200.

I talk about this in the video, but it is possible to do the entire long division process through this level of thinking. For instance, because we know that 100×3 is 300 if we double that 200 × 3 is 600 and that’s going to be too much. We know it’s going to be less than 200. We can place the 100 in the quotient area and subtract 300 from our total to see how many we have left and go from there. If we subtract our 300 from the 582, we’re left with 282. If we multiply 3 by 10 it gives us 30 and we know it’s going to be higher than that, but we know it’s going to be less than 3×100. We can look at different multiples of 10×3 to see how many we can use for 282. We know from our experience that this is going to be 9, so 90×3 gives us 270. If we subtract that 270 from the 282 that we have, we only have 12 left. We know that 3×4 is 12 so we’ve made 100+90+ 4 groups of 3 giving us 194 for our answer.

It is completely acceptable, even encouraged, to explore place value in this way, because it gives the child an understanding of what they are doing. However, they must understand or shift into doing so the place value. For example, looking at the same problem of 582 ÷ 3, the first thing that we do is look at our 500s because we’re going split our 582 by place value giving us 5 hundreds, 8 tens, and two ones. With our 5 hundreds, we can clearly make one group of three using 3 of the hundreds and leaving 2 hundreds. Those 2 hundreds get placed with our tens giving us 28 tens. We can take those 28 tens and make 9 groups of 3. That uses 27 of our tens and leaves 1 ten left over. That ten gets placed with our ones giving us 12 ones, and because 3×4 is 12, we can make four groups of 3 in our ones. That leaves us with a rectangle of 3 on one side and 194 on the other side with 582 total. 

Using Zero to Hold a Place

At some point, we will come to a problem where in our first place value we are not able to have any groups. I think it’s important in the beginning that we go ahead and put a zero in that place value. One it shows that we’re working by place value, and it holds that space. For instance, if you’re looking at 127 ÷ 3, we can’t get any groups of 3 in our hundred, so we put a zero in the hundreds place and move that hundred over to our tens giving us 12 tens.

Using Rounding and Estimation

Eventually, we look at more complicated, long divisions, where we can use rounding and estimation to guess how many we can fit in each place value.  This usually happens when dividing by double digits. Here we use rounding our divisor (number we are dividing by) to give us an idea of what’s most likely to fit in a certain place value. We need to be clear and distinguish between using the rounded divisor to give us an estimation, but then using the actual divisor in our calculations. We are only rounding to give us an estimation, but we’re still using the divisior to find the quotient. I go over this in more detail in the video.

Conclusion

There are many concepts and ideas necessary to understanding and doing long division. It is one of the more challenging concepts of elementary school in my opinion. There is a lot of patience that is needed and a good deal of scaffolding along the way. I hope you enjoy both the video and the post. Please feel to reach out and ask any questions.

Video on Long Division

Time Stamp

00:00 Introduction
00:14 Concepts to Review
02:08 What Division Is
04:25 Area Model Division
08:31 Three-digit Numbers
17:10 Using 0’s to Hold Places
20:00 Divisors Greater than Single Digit
28:21 Long Division Algorithm
31:40 Using Estimation
37:34 Using Rounding
44:10 Closing