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We are having quite a discussion about the language to use when discussing the shift of digits when operated on by powers of ten.
Our interim testing company has a question: What happens to the decimal point when multiplying 15.6 x 10 to the second power. Answers: A. The decimal point moves one place to the left. B. The decimal point moves two places to the left, etc. (you get the idea).
We are asking them to change the language to say the digits shift not “the decimal point moves” because we don’t think the language is precise about what is actually occurring. In the progressions it is discussed that the digits are shifting and not the decimal point. How important is this? Our testing company sites many places where the language of the decimal point is moving and I have also found this. What are your thoughts? Thanks.
What standard might that question be testing? My guess would be 5NBT2 except that standard is about explaining patterns and one instance doesn’t make a pattern.
So maybe the question is more about being familiar with the pattern? Maybe that makes the question more about 5NBT7 and knowing about one piece of the calculation.
Could you elaborate about what the question is supposed to tap?
It seems like 5.NBT.2. Having only one instance presented may not be a problem. If they’ve studied this topic, they should be able apply the pattern in one particular instance. After all, using the pattern in calculations of this sort is one reason to study it.
The language of moving the decimal point is in the standard: “explain patterns in the placement of the decimal point”. If the standard is interpreted in the spirit of mathematics that we should consider inverse problems as closely related, then question like “if the decimal point is moved 2 places to the right, what kind of operation could do that?” be appropriate.
In my observations, describing decimals “moving left and moving right” easily becomes something students memorize and confuse, especially with scientific notation. When converting from scientific to decimal notation, the decimal moves one way and when converting back it moves the other way. The students get it right 50% of the time. That’s not to say we should never say “the decimal moves to the left,” but I have gotten much better results by having students talk about numbers being “one decimal place larger or one decimal place smaller.” Dividing by 10 makes a number one decimal place smaller while multiplying by 10 makes a number one decimal place larger. This connects well with NF.5.b, “Explaining why multiplying a given number by a fraction greater
than 1 results in a product greater than the given number
…; explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b b.Dividing by 10 makes a number one decimal place smaller while multiplying by 10 makes a number one decimal place larger.
Do students ever interpret this as dividing by ten results in one fewer digits after the decimal point (and the other way around)?
Sometimes I think how sad it is that the idea of placing the decimal point under ones lost to the practice we use now. Would make place value names symmetric, emphasize units in a clear way. Oh, well.
I see what you mean and it does matter how we word things. The exercise I give my students involves focusing on relative value. Certainly it is incorrect to state that there are fewer digits because, for example, there may be more digits on the right side of the decimal when dividing by 10. Here is the exercise and maybe this is why I have not been aware of any of my students equating size with number of digits: https://dl.dropboxusercontent.com/u/7405693/Webpg/worksheets%20Alg%201%20SPRING/Scientific%20Notation%20comparisons.doc
I see what you mean. What an interesting thought! It is important how we word things. We do not want them to think dividing 0.05 by 10 will result in fewer digits on the right! I have not observed any students making this wrong conclusion, probably because as we talk about it, we are focusing on relative value. Here’s the quick exercise they complete: https://dl.dropboxusercontent.com/u/7405693/Webpg/worksheets%20Alg%201%20SPRING/Scientific%20Notation%20comparisons.doc
Do you use this exercise in 5th grade? It uses scientific notation, including negative exponents, which currently 8th grade. How do students process negative exponents? Do they memorize a rule like $10^2$ means multiply by 1/100?
I would not use that exact worksheet at 5th grade (right now I use it in Algebra I based on our old curriculum). Here’s a quick, partial modification I envisioned for what you were asking about: https://dl.dropboxusercontent.com/u/7405693/MEGSL/multip%20and%20divide%20by%20ten%20comparisons.docx
In my opinion, it is unwise to imply decimals move as though they have wheels. I would talk about place value and relative size of the number.I can see how in 5th grade a bit longer explanation will be useful. In Algebra I, with some caution, it can be taken for granted, that they know which number is bigger. In 5th grade it’s something that they still have to learn properly (5.NBT.3b). Stating that, for example, 321 is greater, and 0.0321 is smaller, and then using the fact that dividing by 10 makes a number smaller could combine two things that students should work on.
Speaking of “making bigger”. Algebra I students know negative numbers. Do you think that the statement should be “Multiplying a positive number by 10 makes it bigger” so that misconceptions are not reinforced?
I am going to chicken out at this point because I am not sure exactly how the CCSS writers want us to word this with the best (consistent) precision, and Dr. McCallum may want to address it here. In high school, we sometimes say, “larger negative” meaning the number is farther from zero on a number line. Vector magnitude and absolute value fall into this discussion as far as what does it mean to be “larger.” I’m thinking Phil Daro’s video about misconceptions might be helpful because he addresses how a concept understood at one level can best be tweaked at the upper level:
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