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A long time ago, when I was a young teacher tied to whatever textbook we were using that year, I remember a student working on a “word problem” where 30 cookies had to be divided among 8 children. The student pointed out to me that 8 “doesn’t go into” 30, which, if you have a very limited understanding of division, can be thought of as “sort of” true. What this student meant to say was that 8 doesn’t go into 30 evenly, which means there will be a dreaded remainder. So what’s wrong with that?

The problem is this: the people who write word problems involving division don’t understand that 98% of most division problems come out with remainders. So what do we do? We give kids lots of division problems that come out without a remainder, and they expect that this is how 98% of all division problems should come out, and that remainders are the exception.

All of which is not only incorrect, but harmful: if we feed our students a steady diet of division problems that don’t have remainders, they’ll come to believe that they did something “wrong” when they do get a remainder. NOOOOOOOOO! What we should be doing is giving students division problems where most of the problems have remainders, and only the occasional problem should go in without a remainder, which is the most likely scenario.

Not only that, but our students also need to develop a more sophisticated understanding of what to do with remainders: in most cases, we just instruct students to “round up” or “round down,” which is simplistic and fails to stress the importance of interpreting a remainder. In this activity, there are three levels of problems (from 1 to 3 stars), which have different levels of interpretation.

The single star problems (problems 1 - 4) take the same situation and ask students to interpret the remainders in two ways: either the remainder is discarded and only the whole number portion is used (such as how many full saucers of space aliens there will be), and a second problem directly underneath in which the same numbers are used, but the remainder means moving to the next whole number (“how many cartons will I need to hold all the sodas?”)

The next set of problems (5 - 8) are a little more sophisticated: the numbers are bigger, which requires 2 steps in doing the division, and involves using the remainder to answer the question. For example, in question 6, the second part of the problem asks how many more glue sticks will be needed to fill an additional package: to solve this, the student needs to know that the remainder is 3 and that 4 more glue sticks will be needed, so the answer is actually neither the whole number or the remainder. See how cool that is?

The third set of problems asks the student to interpret the remainders at an even higher levels. Students not only have to look at the remainders, but then figure out how many times the remainder goes “into” the divisor. For example, on problem 9, there will be 3 pounds of flour left over, so while the baker has to order 10 sacks of flour on the first day, the following day she only has to order 9, because of the remainder. Do you get how important this is?

How to use this: I like a “fishbowl” approach, which means you can set up 3 containers, each with the different “levels” of the problems, and have students set a goal about how many of each he/she would like to solve. So maybe a student who is struggling with division will do three of the single star problems, and then two of the two star and one of the three star. Maybe another student will focus on doing all fo the three start problems, which take a little more time. It’s up to you.

The problem is this: the people who write word problems involving division don’t understand that 98% of most division problems come out with remainders. So what do we do? We give kids lots of division problems that come out without a remainder, and they expect that this is how 98% of all division problems should come out, and that remainders are the exception.

All of which is not only incorrect, but harmful: if we feed our students a steady diet of division problems that don’t have remainders, they’ll come to believe that they did something “wrong” when they do get a remainder. NOOOOOOOOO! What we should be doing is giving students division problems where most of the problems have remainders, and only the occasional problem should go in without a remainder, which is the most likely scenario.

Not only that, but our students also need to develop a more sophisticated understanding of what to do with remainders: in most cases, we just instruct students to “round up” or “round down,” which is simplistic and fails to stress the importance of interpreting a remainder. In this activity, there are three levels of problems (from 1 to 3 stars), which have different levels of interpretation.

The single star problems (problems 1 - 4) take the same situation and ask students to interpret the remainders in two ways: either the remainder is discarded and only the whole number portion is used (such as how many full saucers of space aliens there will be), and a second problem directly underneath in which the same numbers are used, but the remainder means moving to the next whole number (“how many cartons will I need to hold all the sodas?”)

The next set of problems (5 - 8) are a little more sophisticated: the numbers are bigger, which requires 2 steps in doing the division, and involves using the remainder to answer the question. For example, in question 6, the second part of the problem asks how many more glue sticks will be needed to fill an additional package: to solve this, the student needs to know that the remainder is 3 and that 4 more glue sticks will be needed, so the answer is actually neither the whole number or the remainder. See how cool that is?

The third set of problems asks the student to interpret the remainders at an even higher levels. Students not only have to look at the remainders, but then figure out how many times the remainder goes “into” the divisor. For example, on problem 9, there will be 3 pounds of flour left over, so while the baker has to order 10 sacks of flour on the first day, the following day she only has to order 9, because of the remainder. Do you get how important this is?

How to use this: I like a “fishbowl” approach, which means you can set up 3 containers, each with the different “levels” of the problems, and have students set a goal about how many of each he/she would like to solve. So maybe a student who is struggling with division will do three of the single star problems, and then two of the two star and one of the three star. Maybe another student will focus on doing all fo the three start problems, which take a little more time. It’s up to you.

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15 pages

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