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Crossing the Street This photo is of Wikis Take Manhattan goal code R9, Children/Families, crossing street. |} (Photo credit: Wikipedia)

 One-to-one correspondence.

Materials

  • Children
  • Hands
  • Street to cross

Instructions

  • Ask each of your grandchildren to take your hand when you cross the street
  • Ask them if you have enough hands for each child
  • If you do, ask them to name other families who have enough hands to walk their children across the street
  • Ask them to name other families who do not have enough hands to walk their children across the street
  • Ask them to name other families who have more hands than they need to walk a child across the street.

What Should Happen?

Your grandchildren are identifying sets – parents who have two hands and two children, parents who have two hands and one child, parents who have two hands and more than two children.

Why Is This Important?

A set, in math, is a well-defined group. You know what is in the group and what is outside.

Georg Cantor, a mathematician in the 1800s, invented set theory.  He discovered that you can find out whether one set is bigger or smaller than another set by applying one-to-one correspondence.

Sets (finite, or countable):

  • One parent with two hands and two children
  • One parent with two hands and three children
  • One parent with two hands and one child

There are approximately six billion people in the world. No matter how many parents and children there are, you can count them.

And, you can compare sets. For every parent with two hands and two children, you can match them up to a parent with two hands and one child. Whichever you run out of first is the smaller set.

Cantor applied this insight to infinity, adding and multiplying an infinite number of things.

Sets (infinite, or too many to count):

  • All the whole numbers (1, 2, 3, 4…)
  • All the fractions (1/2, 1/3, 1/4…)

In either case, if you add another number, it will still be an infinite set, so the answer to what is infinity plus one is infinity.

Similarly, the answer to what is infinity times infinity is infinity.

Optional: In answer to the question are there any infinite sets of numbers that do not correspond one-to-one, that is, are different sizes, and if so, how do you tell which is larger?

Yes. The set of real numbers, which includes all the rational numbers, fractions and irrational numbers, square roots, and pi, along a continuous line (1, 1 ½, 2, 2 ¼…) out to infinity, is larger than the infinite set of natural numbers (for counting,  1,2, 3, 4…) and ordering (first, second, third…) because you cannot pair up numbers in the two sets.

The set of real numbers is larger because it has numbers left over when you match them to natural numbers.

Thanks to talkingmathwithkids.com for this activity.

 This post was first published at grandmotherdiaries.com

Carol Covin, Granny-Guru

Author, “Who Gets to Name Grandma? The Wisdom of Mothers and Grandmothers

http://newgrandmas.com

 

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