Estimation: Compensation
by Gail Englert

A response to the question:

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Please give me some techniques for teaching specific estimation strategies, especially COMPENSATION, etc. Thank you very much.

Sandra

Dear Sandra,

I 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 problem.

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 them.

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 T2T service

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