Monday, November 5, 2012

Using the Empty Number Line to Subtract

One of the best bits of advice that I ever got (and now pass on to students, parents and educators) is this:

Good mathematicians look at the numbers first before they decide which strategy to use.
I was in a Grade 4 class recently and the teacher was concerned that she had some students who had trouble doing the traditional algorithm. My concern with the traditional algorithm is not that students shouldn't learn it but they should learn when to use it. This is fast becoming a forgotten skill. Ask yourself if you need to use the traditional method to do the following:
1) 1000-5
2) 1000-998
3) 400-320
4) $10.00 -$2.25
5) 17-8
6) 800-481
7) 5028-2279
I would imagine that most people would do all of these in their heads with the exception possibly of #7 (and maybe #6 too). That being the case, we need to get our students to do so too.
To help students develop this skill, I began by writing the following number string:

I always make the point of writing the question horizontally and not stacking the numbers; it nudges the students into thinking of non-standard approaches. Students wrote their answers on their personal whiteboards and I would say that whilst most just wrote the (correct) answer, there was a core who wrote the algorithm for each question. So I asked the class if it was always necessary to use the algorithm.
Some said yes.
Some said no.
No surprises really. I then asked how they did #5 above. A common method was to subtract the 200 from the 500 to get 300, then subtract the 30 from this to get 270 then subtract the 4 from this to get 266. I represented this on an empty number line:

We then gave students some follow up questions. Some used the empty number line to model their thinking:
Others stuck to the traditional algorithm but it was becoming clear that this was causing issues for them. When attempting 800-481, this student (who at first thought the algorithm was better) had an a-ha moment:
Not successful with the algorithm, but successful using good number sense. He felt pretty good about himself.
There remained a couple of students who insisted that the algorithm was always the best method. One student worked out 100-68 traditionally and when I asked if he could work out 100-60 and 100-70 could only do so with the algorithm:
To me, this is a clear case of when the algorithm is introduced too early it actually undoes number sense.

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