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This lesson was developed at the request of the fifth grade teacher who wanted help teaching long division. Virtually all the students were having difficulty.
Needed Materials: Overhead programmable calculator to do random warm up questions and to generate random numbers for division. (These items can be done by hand and prepared ahead of time.)
Directions:
1. Begin the class by having students practice multiplying by 10, 100, or 1000. I use problems randomly generated by a TI 83 program and displayed with the overhead calculator and viewscreen.
2. Write “division” on the board with a large bubble around it. Ask the students to think about what they know and feel about division. After a couple minutes, have the students think-pair-share with a classmate, then ask for responses and write them around the bubble. Typical “feeling” responses may include “it’s hard” or “it takes too much time”. Tell the class you’re going to look at division today in a way that you hope will make sense to them and that will take much of the guess work out of the task.**
**Note: I believe the concept of division should be developed in this way before taught using the traditional algorithm. However, this activity was in response to a problem that had been brewing for 2 years without resolution. A couple of the students understood the traditional algorithm, but to the majority of the class it was tedious guesswork.
3. Demonstrate first what division is. I used a pile of pinto beans (we pretended they were cookies). I asked what it would mean to divide the “cookies” up among the students sitting in the first row of seats. The students said I would need to split the pile evenly so that everyone got the same amount. (I confirmed that mathematically division means “fair” sharing—that dividing the pile in unequal portions may be “dividing” the pile in plain English, but not in the sense that we use the word mathematically.) I asked the students how to go about dividing the pile of cookies fairly. They suggested counting all the beans, then dividing by the number of students. I told them that would work, but I kidded them that I thought they didn’t like doing division problems! I asked if the “one for you, one for you, …, one for me” method would work. They said it would, so I started divvying out the pile of cookies that way. I made a big deal about getting tired after about 3-4 rounds. I asked them if I could try “three for you, three for you, …, three for me”. Since the pile of beans was still large, they said that should work. After a couple rounds of exaggerated effort, I asked if I could try “ten for you, ten for you, …, ten for me.” They said I might run out of beans. I asked them if that would be a problem if they knew how many beans were in the pile. They said it wouldn’t.
4. Without finishing the cookie pile, I switched to the blackboard. I made up a problem: Suppose we know we have a pile of 157 cookies that we want to share equally with 23 people. How many does each person get? I wrote 157 ¸ 23 on the board (example below). I reminded them to think about our bean experiment. Would we be able to give each person 10 cookies each? No—because 23 sets of 10 is 230—not enough cookies. I asked if we should start divvying the cookies out 1 by 1. Several of the students immediately suggested giving them out 2 by 2. I asked them how many cookies that would take. They replied 46. I looked at the board and remarked that there were definitely enough cookies to do that. I asked them if I did that, how many cookies would remain—what did I need to do to know? They said I needed to subtract the 46 from 157 and that the answer would be 111. I noted that the need to subtract was a unanimous decision, but arriving at the answer was not automatic for many of the students even though the process required no regrouping. (I made a mental note that these students needed to work with subtraction by parts like in “No Need to Borrow”.)
5. We continued with the process. A few of the students wanted to take away 4 23s, but since the majority were still thinking hard about taking away 46, I nodded to the few, gave them the “hold on a second” look, and continued. After subtracting 46 3 times, we were left with 19. Now what do we do? The students were quick to say there weren’t enough to go around again so those were the leftovers. What do we do with them? I offered to keep the extra cookies, but several of the students said we should write that as our remainder. We talked briefly about breaking the cookies in parts and giving people fractions of cookies but decided that that wasn’t practical. Too hard, too many crumbs. I think that reference will be useful, however, when later we need to express remainders as fractions.
6. Finally, I asked how many cookies each person received. We looked at the work and I helped them count the 23s by adding the numbers that are written in red in the example below.
7. We did another example with a randomly generated problem. This time it was 9435 cans of Pepsi to be divided among 87 people. Would it be enough for everyone to have a can a day for a year? A month? What did they think? After some speculation, we began the division process. Stop—estimate—can you give every one 10 cans? Yes, easily, since 87 sets of 10 is only 870. How about 100 cans each? Yes, again, because 87 sets of 100 is 8700. Each person could have more than 100 cans a piece. We tackled the problem as noted below.
8. Finally, we did one “like they write it in the book”. The process is no different. See the third example below.
The mathematical goals are for the student to understand what division is and to use his knowledge of multiples of 10, 100, or 1000 to make long division more accessible.
Comment on the overall effectiveness of this activity. Then use the form below to reflect on the activity from the perspective of both the teacher and the student. Give the lesson points based on how well it went--each question is worth up to 4 points.
This algorithm for division may
be familiar to recent participants in elementary math methods classes, but it
was new to me. I first saw it in
Maurice Burke’s Analysis class (Math 525) at Montana State. I have never seen an elementary teacher use
it. I believe it supports an
understanding of division far better than the traditional left to right
division method. Granted, the
traditional algorithm makes sense to someone who has good number sense, but
typical instruction of the traditional method does not highlight place
value. Instead, most traditional
instruction centers around how to decide what digit (not number) to use as a
multiplier and where to write that digit in the answer. Where the answer is written has little
bearing on the actual understanding of division! Repeated subtraction makes sense and is adaptable to students
entering at various levels of multiplication expertise.
Unfortunately, this activity had
limited success. My first visit to the
classroom was on Tuesday; the next time I was free to come was that
Friday. By Friday, the classroom teacher
had “finished” the chapter on division and was giving the students a test. To the best of my knowledge she did not
reinforce the repeated subtraction method nor assess it on the exam.
I really do believe in this
method! As mentioned in the activity
description, I think it should be used BEFORE teaching the traditional
algorithm. I believe with careful
instruction, students can see how the traditional algorithm is a streamlined
version of the repeated subtraction algorithm.
Then again, is it necessary to teach 2 algorithms in a world where
handheld calculators are going to be used to do most long division?
_4_ Was the purpose of the activity clear to me?
_3_ Was the purpose of the activity clear to my students?
_4_ Did the activity support a range of learning styles?
_2_ Were the students engaged during this activity?
_13_ Total points (out of 16)
The purpose (objective) of the activity was…
I communicated to the students why we were doing the activity by…
…telling the class we were going to look at division in a way that would hopefully make more sense to them than previous methods and that should take most of the guess work out of the task.
The part of the lesson that was student-centered was…
…not much except the group interaction. This was an instructional lesson meant to provide students with a demonstration of division that could be brought to memory as they do division in other contexts. The lesson could be made more student-centered by having students do step 3 individually or with a partner, using a pile of beans and a hypothetical number of recipients. After dividing their beans into equal piles, the group could brainstorm their methods, and then return to doing step 4.
157 ¸ 23 = 6 R 19
- 46 2
111
- 46 2
65
- 46 2
19 done! Count the 23s (recorded in red) and write the answer (also in red)
- 8700 100
735
- 87 1 The students had some trouble doubling 87, so at first we just took away 87.
648
- 174 2 Then I helped them think of 87 as 80 & 7, doubled each, and combined the results.
474 They could see it was going to take a while—I wondered out loud about using ½ of 10 87s
- 435 5
39 done! Count the 87s (recorded in red) and write the answer (also in red)
62 |7065
- 6200 100
865
-620 10
245
-124 2
121
- 62 1
59
This program is written in a “timed” manner so that each problem is visible only a short period of time. There are lots of discussions against using timed activities in elementary classes, and I agree with most of them. However, many students enjoy the challenge of “beating the clock”. They take great pride in being able to answer quickly (and should be able to when multiplying by 10s). I am careful to judge the success level of all the students and slow the program down if even a few are having trouble. It’s easy to speed it up after everyone is experiencing success. Many childhood games focus on doing something slowly at first, then faster and faster. Have you ever sung “John Jacob Jingleheimer Schmidt”? They love it!
[Comments in brackets are for clarification and are not included in the program when entered in the calculator.]
Program:
Multby10 [Random
multiplication by 10s]
:ClrHome
:
:For
(N,1,10) [Loop
that does 10 problems]
:RandInt(0,100) ®A [Randomly selecting
a number]
:RandInt(1,3) ®B [Randomly
selecting the power of 10]
:
:Output(1,1,N) [Prints the
problem number on the screen]
:Output(1,3,”)”)
:
:Output(4,7,A) [Prints
the problem on the screen]
:Output(4,10,”*”)
:Output(4,13,10^B)
:
:For(M,1,1500) [Timing
device: change the 1500]
:End [to
increase or decrease time allowed]
:
:ClrHome
:End [End
of 10 problems loop]