Thursday, February 21, 2008

4.58 x 10,000 = 4.580000

Most of my numeracy students remember that helpful rule from middle school: "Multiplying by 10 means adding a zero", and so we get results like the title of this post. This is one of those fundamental place-value problems, the type of thing that betrays just how little some students really get about the number system. It's taken about two weeks of practice to get them comfortable with the idea of shifting the decimal place left and right (and remembering which way to shift it, depending on the operation).

We are also currently struggling with the issue of the missing decimal point... when there is no point shown in a number, where is it really? Some of my students still think that you put the point at the front of the number. Why do they think this? I'm not sure. Before break, we spent a whole lesson on what the decimal point means, and it seemed to go well. Since we've been back in the second semester, the question of where the missing decimal point goes has been asked and answered many times each class period. They are getting better at comparisons: if I ask them to compare 473 and .473, or .4 and .39, or .4 and .04, they are usually getting it right. And yet, when faced with the problem 473 ÷ 100,000, some students seem to forget it all and start with the decimal at the front of the 473 (or sometimes between the 4 and the 7), forgetting that this changes the value of the number.

No wonder scientific notation is such a bear to teach in Algebra... To reinforce both concepts, I've been teaching scientific notation (with positive exponents only) in this unit, and it's finally starting to work. From the start, my students could tell me that 10^6 was the number 1 followed by 6 zeros, but they couldn't see the relationship between the problems 9.02 x 10^6 (which was totally confusing) and 9.02 x 1,000,000 (which is finally becoming easy). Converting a number into scientific notation is starting to make more sense to them now, since I've finally figured out another flaw in some of the students' understanding: they don't really get the significance of the equals sign. I would show over and over why 302,000,000 = 3.02 x 10^8, and some kids just weren't catching on. But then, when I asked them what they would get if they multiplied 3.02 x 10^8, they were surprised to see that it was 302,000,000. I would get lots of "ohhhs" as they realized that the two parts of the equation had to be the same, and that you could multiply to check your answer. The main problem I still have is getting them to remember that the first part must be between 1 and 10. But at least we're making progress! Though we have been learning dividing by powers of 10 at the same time, I don't want to introduce scientific notation with negative exponents now (since they have never seen negative exponents before). I want to give this time to sink in, and maybe come back to it later in the year.

We have the rest of this week off for winter break; when we start next week, I think it's time to move on from this percent and decimal concepts unit and start in on fraction operations. 1/2 + 2/3 = 3/5, here we come! (One of my favorite things to show numeracy students is why this equation doesn't make sense.)


Mr K. said...

It's taken about two weeks of practice to get them comfortable with the idea of shifting the decimal place left and right

This is one of the few times a year that I bust out the calculators. It quickly lets them figure out both the moving the decimal point aspect, and the adding a zero aspect. They end up not thinking of it as adding or taking away zeroes, but as moving the decimal point (which I'm not completely happy with, but I'm having to live with) and filling in with zeroes to take up space as needed.

My kids also learn that the exponent doesn't mean how many zeroes you add, but how often you need to multiply by (or divide, in the case of a negative) the base. It seems like a trivial difference, but once they get the "times ten means move the decimal" from the calculator exercises, it's easy to build on just repeating that.

I did some reflections on this series of lessons a couple of weeks ago, which went much better than I expected them to.

Dan Greene said...

I know what you mean.. The last time I tried this unit, we used base-10 blocks and exchange rates, and saw that multiplying by 10 means shifting all the digits one place to the left, and filling in zeros for any places left empty. But, in practice, of course, we see it as moving the decimal point to the right, not the digits to the left. This confused them completely (especially when division came into play as well), and we made little progress. At least this time around, they are more able to actually do the problems. (I feel like a sell-out :)

Some students still forget which way to move the decimal for multiplication or division... I ask them to think about whether the number should be getting bigger or smaller considering the operation they are using, and this usually helps. Though I know it will come back to bite me when I teach fraction multiplication, and suddenly certain multiplications cause the result to be smaller instead of bigger! And divisions that will leave you with a bigger quotient!

I did see your posts, by the way. I like the look and feel of your worksheets a lot.