Activities 4 and 5 Findings

Working with Children on Activities 4 and 5

They did it!

Roger (11):

Yes. I would say that ToonTalk can count all fractions every one even though there is lots.

Susan and Tom (13) wrote:

Yes. If we got all the fractions, we could spend forever counting them but it would be possible.

Susan and Tom said that the output of their robots was countable since you can take the first fraction off the nest and say “one”, then the next and say “two”, and so on.

Ivan had planned to modify the robots that generate all the proper fractions to produce the rational numbers greater than 1 but then he said:

So, I'll use the team of robots generating the proper fractions to feed my new robot. I will divide 1 by each of the coming numbers and give the result to a new bird.

Sally says that something is fishy here. She says:

“There must be the same number of rational numbers from 0 to 1 as there are from 1 to 2. And from 2 to 3. And 3 to 4. So there are an infinite number of intervals of the same size as 0 to 1. So there must be an infinite times more rational numbers greater than 1 than between 0 and 1.”

Do you agree or disagree with Sally? Why?

Susan and Tom (13) wrote:

Tom says he agrees. [My answer is] Sort of but you can’t have ∞x∞ so what is she [Sally] going on about.

Susan then suggested starting with the lowest number between 0 and 1 and have it dance with the lowest number greater than 1 and so on. Tom asked her how she could get the smallest number like that. She saw this was problematic and then she suggested they start with those with 1 decimal place, then those with 2 decimal places, and so on.

The researcher said that was a fine way to get the numbers between 0 and 1 dancing with those between 1 and 2 but what about all the numbers greater than 1. Susan said you could do that too and started a complex explanation until Tom pointed out they could use the sequences they just generated.

Ivan (12-13) said

Sounds reasonable... but...

So you want me to find a fraction in the interval [0, 1] for each fraction greater than 1 which Sally would choose. In other words, we could combine each number of Sally (>1) with a number <1. That does not seem very difficult, e.g. if she chooses 205/3 I'll answer with 3/205, if she chooses 101/11, I'll take 11/101, and so on.