Principle of Green Tick Number Line Induction

The principle of mathematical induction is the hardest to understand particularly for students in lower grades.  However, the color green is easy to understand.  Green also can be used in sentences in a way that is compact and concise yet transparent in meaning.

Consider a number line.  The ticks on the number line are whole numbers, i.e. natural numbers, starting from 0.  So 0, 1, 2, etc.

Suppose that 0 is green.

Suppose each green tick is immediately followed by a green tick.

Then all the ticks on the number line are green.

Variations of this can be given to emphasize certain points.

If 0 is green, and if the tick after a green tick is also green, then all the ticks are green.

If 0 is green, and if the whole number tick after a green tick is also green, then all the whole number ticks are green.

This is understandable to children.  As soon as they understand the number line, they can understand ticks on the number line being green.

Green means go. So it is a good color to use for this purpose.  We can also talk about green sets or green sets of ticks, or sets of green ticks.    This can transition in set theory.

Previously,

http://newmathdoneright.com/2012/05/14/peano-axioms-number-line/

  1. Zero is a unique tick on the number line.
  2. For each tick on the number line, there exists a unique tick immediately to the right of it.
  3. Zero is not a tick to the right of another tick.
  4. If the ticks to the right of two ticks are equal, then said two ticks are equal.
  5. If a set contains zero and each tick to the right of a tick, then it contains all the ticks on the number line.

Now becomes

  1. Zero is a unique tick on the number line.
  2. For each tick on the number line, there exists a unique tick immediately to the right of it.
  3. Zero is not a tick to the right of another tick.
  4. If the ticks to the right of two ticks are equal, then said two ticks are equal.
  5. If the zero tick is green and each tick to the right of a green tick is green, then all the ticks are green.

Being the color green means belonging to the set.  We can have different sets have different colors.

In a development of this color scheme, consider the following.  Fractions can be yellow.  Infinite decimals that are not fractions can be red.  Other colors can be used as well.

An infinite decimal is a greatest lower bound or least upper bound of a set of fractions. So using red for these numbers relates to their being a bound.  Red means stop.  So a set of fractions stops at the least upper bound.  So square root of 2 is a red number.  The fractions whose square is less than 2 stop at the square root of 2.

An alternative is that any number that is a least upper bound is red, including fractions and whole numbers.  I.e. is a stopping number.  A stopping number is the same as a least upper bound.

We can use green for whole, yellow for fractions and blue for infinite decimals not a fraction.  Then red can be added as stopping number, producing mixed colors, so that red and blue is purple. So the decimals not fractions are purple. This signifies they are stopping numbers and not rationals.

==

A set can be thought of as a team.  Being on a team is signified by a color or colors. So the green team is one team or one set.

If 0 is green and each green tick is followed immediately by a green tick, then all the ticks are green.

This is easy to follow if you are used to thinking of the green team, blue team, etc.

=

Because Peano Axioms are the logical basis of whole numbers, and because set theory is the logical basis of teams, we can use words and terms that students are familiar with to get across the message of the Peano Axioms and set theory.

The meaning of teams and sets is related because teams are a type of set.  Colors to signify membership of a team can be adapted to mean membership of a set.

When we use math foundations, we will find such tricks come naturally.  If we go with the true meaning of numbers and math objects from math foundations, then we will find that we can adopt terminology and sentence structure from applications like teams to make the set theory and abstract number concepts easier to understand and remember.

This will build on itself as people use the Dedekind version of New Math.  Dedekind thought through the logic of numbers better than anyone before or even since.  So going back to his approach is the way to the most logical and shortest path through arithmetic.

This contrasts to math democracy where every idea is presented as equally important or structurally significant even if it is an application or mixes concepts like fractions and length.  Mixing concepts without realizing it is extremely harmful to students progress and will bring it to a halt.  It means learning becomes rote procedural and not conceptual.

Advertisements

About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
This entry was posted in Uncategorized. Bookmark the permalink.

One Response to Principle of Green Tick Number Line Induction

  1. Pingback: The Principle of Math Induction is Common Sense | New Math Done Right

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s