Train a robot called
Doubler that receives a number on a nest and
gives a bird twice that number. Illustrated instructions for training this
robot are the end of this handout. Test out
Doubler by giving it a box like this:
Give the
In bird a number and watch what
appears on the
Out nest. Give the
In bird a different number. If
Doubler is working as it should then
whatever number you give the
In bird you should get twice that
number on the
Out nest.
Clear off the
Out nest of all the numbers. Give the
In bird 1, and then 2, 3, 4, and 5.
What numbers appear on the
Out nest?
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Let's give
Doubler all the natural
numbers (1, 2, 3, and so on forever). Since we can't type in all the
natural numbers we'll need a robot to help us. We'll call him
Add 1 since that is what he'll do over and over again.
You'll find instructions for how to build
Add
1 at the end of this handout.
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To test out
Add 1 give it a box like this:
and watch what numbers end up on the
Nat
Nums nest.
Try connecting
Doubler to the sequence of natural numbers made by the
Add
1 robot. You can do this by giving
Doubler the box:
Since the
In hole is empty the robot won't do
anything yet. Give the
Add 1 robot this box:
To get these robots to work together take out a nest with an egg in it.
Give it a nice name. Drop the bird in the
Numbers hole and the nest in the
In hole.
Did you notice that the first numbers coming out are
even?
Will there ever be an odd number?
Explain
Train a robot called
Split. As numbers arrive on the
Input nest
Split gives them alternatively to bird
A and
B. Instructions for training
Split are at the end of this handout.
Put the nest that receives the natural
numbers produced by the
Add 1 robot in the hole labeled
Input and give the box to
Split.
Give the Add 1 robot this box:
What kind of numbers end up in
A's nest?
What kind of numbers end up in
B's nest?
Copy the nest that will receive all the
natural numbers produced by the
Add 1 robot. Start up the
Add 1 robot and you should see
each copy of the nest receiving 1, 2, 3 and so on.
Start again and give one copy to
your
Doubler
robot and the other to the
Split robot. Now start up the
Add
1 robot
again with the box:
If all the robots ran forever would there be a difference
between the numbers that are on
B's nest and the output nest of
Doubler?
If there is a difference, what is it?
If there is no difference, how can they be the same
sequence if
Doubler uses every natural number and
Split
puts only half of the numbers on
B's nest?
If you saw a room full of boys and girls and everyone
was dancing with a partner of the opposite sex, could you figure out if there
was the same number of boys as girls? Without counting!
How would you convince someone you are right?
Imagine that you are running the
Doubler robot and that after the
Add 1 robot gives a bird a number it is copied and begins dancing with the
output number of the
Doubler robot. So 1 dances with 2, 2 with
4, 3 with 6, and so on.
If this continued forever would every even number be
dancing with a natural number?
Would every natural
number be dancing with an even number?
What does this tell you about how many even numbers
there are?
You may be confused at this point. Infinity confused
some of the world's best mathematicians. Galileo Galilei wrote in 1638 about
how one line of reasoning leads you to think the sequences have the same number
of numbers while another line of reasoning leads you to think there are fewer
even numbers.
What do you think?
If you think they all pair up then how do you explain
that the sequence of natural numbers
contains all the even and all the odd numbers?
If you don't think they pair up, what is different
from the room of boys and girls?
Activity 1,
Task A:'
How to Train
the
Doubler
Robot
Activity 1,
Task A:'
How to Train
the Add 1 Robot
Activity 1,
Task B: 'How to Train the
Split Robot
