# Number Systems Unit

Time: The unit will last three weeks, each of which consists of four 70-minute classes.  This year, this means the unit will begin on September 13th and conclude on October 1st.

Main resource: Dr. Rachel Hall’s class page for Multicultural Mathematics is an immensely useful and practical resource. Expect much of my material to come from this site.  Other resources I’ve found so far seem trivial by comparison — if you know of something decent, please share in the comments.

ESSENTIAL QUESTIONS.

How did different civilizations write numbers?

What are the different types of number systems in use?

Why are different bases in use today?

STANDARDS.

Skills that I can assess (i.e. SBG concept checklist items)

1. Translate numbers from non-Hindu-Arabic systems into decimal form and back [first in Mayan, then Egyptian, then others]
2. Perform operations on numbers in alternate bases [first in Mayan, then Egyptian, then others]
3. Determine patterns in a sequence of numbers by performing operations on successive pairs
4. Change numbers from base 10 to a different base, and vice-versa [specifically base 20, base 2, base 60, base 12, and their “project base”]
5. Explore alternate multiplication algorithms (Mayan, Egyptian, and possibly Gelosia [lattice] multiplication or Vedic multiplication)
6. ??? Determine primary and secondary bases of other number systems (c.f. Quiz 2) [example of secondary base: role of 5 in Roman Numerals]

ASSESSMENTS.

• SBG #1 – Friday, Sept. 17. — Concepts 1, 2, and 3 (Mayan only)
• SBG #2 – Thursday, Sept. 23. — Concepts 1, 2, 3, and 4 (Mayan, Egyptian, binary)
• SBG #3 – Thursday, Sept. 30. — Make all concepts available

Project: Number systems project, planned to be very similar to one on the SJU site. [the following has been copy-pasted from this site, and there will be adjustments]

Due Friday, October 1st, 2010

Projects should be about 5 pages typewritten or neatly handwritten, including diagrams.  You may do the projects in groups of 3 or 4.  Ninety percent of your grade will be the overall grade I give the project;  ten percent will be based on a peer evaluation.  Students who do not participate in the project will receive a zero for the assignment.  Projects will be graded on completeness, mathematical content, and creativity.

Fantasy Math. Create your own number system, as J.R.R. Tolkein did in the Lord of the Rings. You’ll need

• spoken names for your numbers
• a positional written system for your numbers
• a multiplication table for 1x 1 through 12 x 12 (that is, however you write 12ten in your system), plus comments on interesting patterns in the table
• a translation of the numbers 100, 157, 517, and 1000 into your number system
• a sketch or model of a mathematical “artifact” produced by your fictional culture
• a history for your number system, including a description of the (fictional) culture that produced it and the reason why the base was chosen

The only restriction is that your system can’t be base 10, 12, 5, or 20, since we’ve already studied examples like these.  I recommend that you use a base smaller than 12. IMPORTANT:  Do not split the project up into pieces, with each person doing only one piece.  It is imperative that everyone in the group is familiar with the number system.  I will give you some class time to work on the project;  however, you should also schedule meetings outside of class.

Relevant NCTM standards

Number and operations:

• develop a deeper understanding of very large and very small numbers and of various representations of them;
• use number-theory arguments to justify relationships involving whole numbers. [This might fit in with some nice problems to solve, such as Egyptian fractions and the unsolved Erdős conjecture involving numbers of the form $4/n$]

Communication

• organize and consolidate their mathematical thinking through communication;
• communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
• analyze and evaluate the mathematical thinking and strategies of others;
• use the language of mathematics to express mathematical ideas precisely.

Connections

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

Representations

• create and use representations to organize, record, and communicate mathematical ideas;
• select, apply, and translate among mathematical representations to solve problems;
• use representations to model and interpret physical, social, and mathematical phenomena.