I was going to move on from arithmetic to algebra, but the discussion of long division led me to think about some of the more ordinary difficulties students have asked about in that area. Here I will show several questions about the process of long division in which zero caused trouble.
Zero in the quotient
When you come across a step in long division where the divisor is larger than the current working dividend, you need to put a zero in the quotient. Students often have trouble here:
Problems with Zero; Writing Remainders When I do long division I have trouble with some of my answers, where the answer is correct except for one thing in the middle that sometimes contains a zero. For example, when I did this one: 38/1294058 I got the answer 3465 r23/28 but when I checked the answer it read 34065 r23/38 Where am I going wrong?
Can you see three different issues here to talk about? The main one will be the zero in the quotient that Lacy missed; but there is also her (or the book’s) use of “r” with a fraction rather than a remainder, and writing 38/1294058 (the format used in doing the division) where we would properly write 1294058/38, or 1294058÷38. In addition, the answer given is not correct! I couldn’t find a likely correction, so I just went with the problem as given. (That was fine, because in doing this I wasn’t giving anything away: Lacy already knew the correct answer, and evidently the problem I did wasn’t the one she had to do, anyway!).
You probably just forgot to write a zero in the quotient when you had to "bring down" an extra digit from the dividend because the remainder was bigger than the divisor. I'll do the problem in a way that avoids that: ____34054_r_6 38 ) 1294058 114 --- 154 152 --- 20 0 <-- the problem is here -- 205 190 --- 158 152 --- 6 (I don't know where one of us copied wrong, but I'll go with this problem anyway!)
What I did here is to follow the standard process carefully, actually writing down the product, 0, where the more efficient method is to skip that step and just write a 0 in the quotient:
____34054_r_6 38 ) 1294058 114 --- 154 152 --- 205 <-- two digits brought down need two digits 190 in quotient --- 158 152 --- 6
Writing a little more, taking it slowly, is all that is needed here. Like last time, remember that correctness must be developed before efficiency. (At the end, I did point out the other issues in the problem.)
Zero in the quotient, after a decimal point
A mother sent us this division question in 2006:
Adding Zeros in a Long Division Problem I was trying to remember the long division rules when dividing a small number such as 6 by a larger number like 351. I am teaching my daughter and I want to make sure I understand the rules about adding zeros correctly.
Zeros come into play twice here. We have to add zeros to the dividend to add decimal places; and for a number as small as this, we will have zeros at the start of the quotient. Dr. Rick just went slowly through the process, paying special attention to how to organize the work:
I prefer not to pile up too many rules, but to help a student see that what you do in this case is exactly what you always do in division. There are really no new rules; the only thing is to recall that the numbers 6, 6.0, 6.00, 6.000, and so on are all names for the same number. I start by looking at the division problem ___ 351 ) 6 How many times does 351 go into 6? None--that's 0, so I put 0 in the quotient, above the ones digit of the dividend, multiply 0 by 351, and subtract this from 6. 0 ___ 351 ) 6 -0 -- 6 Then I'm supposed to bring down the next digit. There is no next digit ... but there is, if I write 6 as 6.0. When I add the decimal point to the dividend, I also put one in the quotient directly above it. 0. _____ 351 ) 6.0 -0 -- 6 0 Now (since I ignore the decimal point at this stage) I'm trying to divide 60 by 351, and 60 still isn't big enough; 60 divided by 351 is 0 again, so ... 0.0 _____ 351 ) 6.0 -0 -- 6 0 - 0 --- 6 0 It's time to bring down the next digit of the dividend again, but again there isn't one ... unless I write the dividend as 6.00:
… and so on. Note again the value of not trying to go too fast. Once everything makes sense, then we can look for ways to improve the process:
Once a child understands how this works, just following the basic rules, I can show some secrets that make it easier! You don't have to write so much, because of what you know about how numbers work. Whenever the quotient digit is zero, you know that when you multiply zero by the divisor, you'll get zero; and you know that subtracting 0 from any number leaves it just the same. You can do those operations in your head, because they amount to doing nothing! All you need to write down is this: 0.0 1 7 _________ 351 ) 6.0 0 0 -3 5 1 ----- 2 4 9 0 -2 4 5 7 ------- 3 3 The only danger in leaving out those steps is that you might lose track of where the next digit goes in the quotient. That's why I emphasize that the digit goes directly above the ones digit of the partial dividend. If you find that this leaves a blank to the left of the digit you add, it means that you forgot to write down a zero; so put it in.
Here’s another example of the same kind of problem, but the tutor who wrote us had hit a roadblock with the student:
Decimal Multiplication and Division I'm tutoring a 6th-grade student, and I'm having a hard time explaining to her why you sometimes need to put a zero in the tenths place, in the quotient - as a sort of placeholder, I tell her. For example, we were dividing 0.5 into 0.025. She's okay with multiplying both numbers by 10, so that the problem becomes 0.25/5, but when we start writing digits for the quotient, she won't believe that the answer is 0.05. She insists that it must be 0.5. I've tried showing her that 0.5 x 5 = 2.5, and not 0.25. And I've tried using 25 base 10 cubes to represent 25 one-hundredths, and then having her divide them into five groups, each of five one-hundredths. She still says you cannot put that zero in after the decimal point. It seems like something is just missing. For one thing, she has it firmly in mind that her teacher said to never add a zero after the decimal point, so she refuses to believe any of my explanations. (For another, she's still a bit shaky on decimal multiplication.) Do you have any other way of explaining decimal division when the numbers that are not strictly analogous to regular division - i.e., when you have to put in a zero in the tenths place, or hundredths place, etc.? (I suspect my student is not the only one who could use help in this area.)
I took Susan through several ways to help the student overcome her “phobia about zeros” and see that meaning trumps memorized rules.
It sounds as if the student is stuck on rules and doesn't dare try to understand what's going on. The claim that "you cannot put that zero in after the decimal point" must come from something the teacher said in some other context, and she feels math is all about rules. That probably happens more often than we'd like to think; too many teachers emphasize rules rather than reasons. So let's first see where that rule might come from, and maybe we can untangle it from this problem. All I can think of is just that once you have a number, you can't insert a zero between the decimal point and other digits without changing its value. You'll have to ask her to give an example where her teacher might have used that rule, and then show why it doesn't apply here. I explain decimal multiplication and division in terms of fractions (assuming a student has an understanding of fractions!). In this case, 25 5 25 10 25 * 10 10 25 0.025 / 0.5 = ---- / -- = ---- * -- = -------- = ---- * -- 1000 10 1000 5 1000 * 5 1000 5 So the answer will be 1/100 of 5, or 0.05. This will probably be too complicated for your student. Here's an alternative way to show the same thing: 0.025 1000 * 0.025 25 25 1 ----- = ------------ = --- = -- * --- 0.5 1000 * 0.5 500 5 100
Susan said she would try my ideas out, but I never heard back about the results. The most important thing would be to break the dependence on rules. Misremembered rules are worse than no rules at all!