Q&A #4319

Estimation: Order of Magnitude, Compensation

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From: Gail (for Teacher2Teacher Service)
Date: Jul 24, 2000 at 10:42:01
Subject: Re: Estimation: Order of Magnitude, Compensation

I don't know what is meant by "order of magnitude", but I do teach my fifth
grade students about compensation, which is really a way of adjusting the
addends to make the addition problem easier to solve.  This topic is
presented near the beginning of our math textbook.  It is in the estimation
section, but I don't think it is really an estimation activity, because it
involves finding the exact answer.  Here is how it works...

What you do is adjust the numbers being added or subtracted to make the
problem easier to do mentally...

for example, if the problem is 198 + 217, you could take 2 from the 217 and
add it to the 198.  Now you would have 200 + 215, an easier mental math

If the problem was 217 - 198, you could add 2 to BOTH numbers to get 219 -
200, again, easier to solve mentally.

To help my students understand why this works, I give them a set of problems,
for example,
191 + 225
192 + 224
193 + 223
194 + 222
195 + 221
196 + 220
197 + 219
198 + 218
199 + 217, etc.  but I don't arrange the problems in that order.  Instead I
scramble the problems up, and then give my students a few minutes to solve

We discuss what we found.  It doesn't take long for the students to recognize
that all the sums are going to be the same...   then I guide them to figure
out why it is happening.

I do this by rearranging the problems in order, as they are in the list
you see above.  Then I let students discuss what they notice, guiding them
to see the pattern.

Once we have that pattern, I give them one subtraction problem and ask them
to generate a set of subtraction problems that all have the same difference,
and we use their sets to determine how the process works for subtraction.

So, if I give them 251 - 139, they might tell me that these problems have the
same difference
250 - 138
252 - 140
253 - 141
254 - 142
255 - 143
256 - 144
257 - 145
258 - 146
259 - 147
260 - 148
261 - 149

Of course, they wouldn't tell me this list in the order I have given it here.
As the teacher, it would be up to me to help students find a way to organize
this inforamtion so it is easier to draw conclusions from.

From that point it is a small step to applying it to other problems.
Every student won't want to use this method, but it really helps those who
are still having trouble regrouping when they add and subtract.  I have
students who will use it every once in a while, long after we have left the
topic, and the others will recall the lessons.  I think it is a valuable tool
for students who are having difficulty regrouping, and it can liven up a
computational practice lesson.

 -Gail, for the Teacher2Teacher service

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