I'd go for neither. They aren't subsets of each other, they are different ways of representing quantities. The same quantities are being represented regardless whether written as a fraction or a decimal. Any number that can't be written as fraction (rational) isn't any better represented by a decimal approximation as it is irrational and therefore representative of a very specific relationship where another representation or notation would be more appropriate. It's important to remember that our symbolic number system is an abstraction to represent quantities and relationships and there are often multiple representations of the same quantities, but also times when only one representation is suitable.

Tried this one out on my grade 11s today. Somewhat as I expected, most got to the counterexample of irrationals as being the sticking point, but very few could articulate how that altered their understanding of the size of the "sets". Depressingly a large portion of the class initially argued that "Decimals must be a subset of fractions" because you always need a fraction to get a decimal (insert face palm here) though with prompting questions a number of them wanted to change their minds. A few students were able to identify that they are just different representations, in their words, "they're the same thing, parts of a whole" so there must be the same amount of them. This was probably the best of the responses. Thanks for the idea, it was a pretty good discussion.

Hmm...I actually agree with the large portion of the class. I'd say "decimals are a (proper) subset of fractions" What is a decimal? I'd define it as a number that can be written in decimal notation. What is decimal notation? It's notation for writing fractions whose denominators are a power of 10. In this notation the number without the decimal point represents the numerator of our fraction and the location of the decimal point (the number of positions from the right of the number) represents the power of 10 in the denominator. So only fractions that can be written so that the denominator is a power of 10 are decimals. It is an interesting challenge for students to try to determine which fractions that is.

Some may try to extend decimal notation to include situations where the number without the decimal point has infinitely many places and the decimal point is thus infinitely many places from the right, but this isn't so easy to do and isn't really appropriate until more advanced concepts like infinite series are covered.

I'd go for neither. They aren't subsets of each other, they are different ways of representing quantities. The same quantities are being represented regardless whether written as a fraction or a decimal. Any number that can't be written as fraction (rational) isn't any better represented by a decimal approximation as it is irrational and therefore representative of a very specific relationship where another representation or notation would be more appropriate. It's important to remember that our symbolic number system is an abstraction to represent quantities and relationships and there are often multiple representations of the same quantities, but also times when only one representation is suitable.

ReplyDeleteTwo thoughts to torture your kids with:

ReplyDeletepi/2?

The Full set is a subset of itself..

Tried this one out on my grade 11s today. Somewhat as I expected, most got to the counterexample of irrationals as being the sticking point, but very few could articulate how that altered their understanding of the size of the "sets". Depressingly a large portion of the class initially argued that "Decimals must be a subset of fractions" because you always need a fraction to get a decimal (insert face palm here) though with prompting questions a number of them wanted to change their minds. A few students were able to identify that they are just different representations, in their words, "they're the same thing, parts of a whole" so there must be the same amount of them. This was probably the best of the responses. Thanks for the idea, it was a pretty good discussion.

ReplyDeleteHmm...I actually agree with the large portion of the class. I'd say "decimals are a (proper) subset of fractions" What is a decimal? I'd define it as a number that can be written in decimal notation. What is decimal notation? It's notation for writing fractions whose denominators are a power of 10. In this notation the number without the decimal point represents the numerator of our fraction and the location of the decimal point (the number of positions from the right of the number) represents the power of 10 in the denominator. So only fractions that can be written so that the denominator is a power of 10 are decimals. It is an interesting challenge for students to try to determine which fractions that is.

ReplyDeleteSome may try to extend decimal notation to include situations where the number without the decimal point has infinitely many places and the decimal point is thus infinitely many places from the right, but this isn't so easy to do and isn't really appropriate until more advanced concepts like infinite series are covered.