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# Five Rules to Draw Number Lines Introduction

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This free version is a short version. The full version will not have the Introduction in the title, but will include the rest of the worksheets in this first package.

These worksheets teach 5 rules to draw number lines. These 5 rules are

Rule 1. 0 is a node.

Rule 2. There is a single arrow out of each node.

Rule 3. There is no arrow into node 0.

Rule 4. Only one arrow goes into a node.

Rule 5. There is a single path of all nodes and arrows.

A node is a point and corresponds to a natural number. An arrow is an ordered pair of nodes, or of natural numbers. An arrow is actually the basic unit of a function. So the worksheets teach the function concept in disguise in terms of arrows. It also teaches sets in a less abstract way as a path of nodes.

These rules correspond to the Peano Axioms which are the foundation of both arithmetic and school algebra.

This is the first installment in a multipart series that will teach arithmetic and algebra with few logical gaps or skips. This will be done in terms of drawing instructions to the extent possible. This will make it an activity and make it understandable at much lower age levels.

The difficulty of most of the graphs given is no more than kindergarten level. The few that are slightly more difficult are still not hard and can serve as a challenge for students who need one.

Most people looking at this will have the reaction of why bother to teach such trivial rules, they are all obvious. This is the goal and art of these drawing instructions. They reduce the Peano Axioms which sound hard to 5 rules to draw number lines which sound trivial.

As we go through kindergarten through 3rd grade math, we will use these same 5 rules for drawing number lines over and over to teach order of whole numbers, counting cows by one to one correspondence, addition, multiplication, subtraction and division.

As this goes on, the student will recognize each of these seemingly different topics as applying the same 5 rules of drawing number lines over and over.

This is common core math without the logical gaps. Arithmetic without logical gaps was developed as the New Math of the 19th century of Grassmann (1861), Dedekind (1888) and Peano (1889). 1960s New Math and Common Core are really Euler's 1765 Algebra book with a few words of 19th century math sprinkled in,
but not the logical order and connectedness that was developed principally by Dedekind in 1888. Dedekind showed how a series of logical baby steps could go from simple set theory and counting rules to arithmetic and order of whole numbers.

1960s New Math and Common Core leave out most of these baby steps. This leaves logical gaps. Because of this, Common Core seems like a series of random lessons that are not connected. This denies students the aha feeling of understanding that the baby steps in logical order gives them.

Using drawing instructions as a way to teach the Peano Axioms makes it concrete and active so that actual students will be able to find this thread as they work through successive worksheet packages.

By using drawing instructions, the rules and the exercises are robust so that as teachers and students restate them in their own words, their logic is preserved. This is key to teaching the abstract concepts in common core math.

There is more information at the website newmathdoneright.com and in a series of e-books at well known e-book vendors. These present more examples and worked problems for the Peano Axioms than any other source. They also have more detailed discussion in paragraph descriptions than any other source.

This material can be used at varying grade levels. It is not more difficult than other kindergarten level material with some help from the teacher. Higher grade levels will be able to read the material with little assistance.

Students in remedial math will find this material of particular value. They can get stuck in the logical gaps that fill standard math curricula. By starting from the Peano Axioms, and using them in baby steps to do each task in arithmetic, the logical gaps are filled in. Some students can learn math by rote, but others get blocked by logical gaps and can't go on. This material will help those students.

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9 pages
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