PEMSTL Mid- winter Institute January 10, 2009

Place the numbers 1-8 in the rectangles so that no two consecutive numbers are next to each other horizontally, vertically, or diagonally.

Powerful Ideas in Mathematics

COMPOSITION In order to do mathematics, the human mind must compose units, which are countable objects, and the conception of units must be flexible. The act of composing units if referred to as composition

COMPOSITION Counting Units of measure Clustering units What constitutes a shape? Operations – adding, multiplying Activity: Geometric Composition

DECOMPOSITION In order to do mathematics, the human mind must be able to decompose units into smaller pieces, The act of forming smaller pieces from units is referred to as decomposition.

DECOMPOSITION Break into parts Fractions Decimals Ratio Percent Measurement Operations – subtracting, dividing Discuss how to complete the following problem: 56-29

RELATIONSHIPS In order to do mathematics, the human mind must perceive of relationships between units and/or partitions of units as entities that can be studied, describes, and manipulated.

RELATIONSHIPS Between numbers and sets of numbers Between shapes and parts of shapes Ratio Proportion Scale Statistics Probability Functions I was traveling to Logan. I passed a sign that said Logan was 48 miles away. My speedometer read 70 mph. My car typically gets 24 mpg. What relationships can be discussed?

REPRESENTATION In order to do mathematics, the human mind must conceive of ways to represent abstractions with some form of symbols that can be manipulated and upon which operations can be carried out in proxy.

REPRESENTATION Written symbol or drawing stands for an idea Numerals Symbols of language that makes sense “It is never enough in mathematics to simply learn the symbols and the rules that govern their use. The symbols are only representations for complex ideas, and it takes time and effort to fully explore those complex ideas.”Schwartz, 2008

CONTEXT In order for mathematics to be meaningful, it helps to have a context in which the mathematical ideas reside. In most cases, real- world applications for mathematics provide the necessary context.

CONTEXT See the real-world practical context from which mathematical abstractions are derived Problem-based learning Why should students invest the effort to learn? Teachers need to know the use of the mathematics they are teaching Discuss: How can you make learning multiplication and division facts relevant to students?

NUMBER SYSTEMS

Number Systems Unury system – one mark for each number Mayan - base 20 Sign value notation Egyptian – base 10, hieroglyphics Roman numerals Place value notation Hindu-Arabic

Bases Binary – 0 and 1, based on vacuum tubes being open or closed 5 (quinary) 8 (octal) – used by the Yuki tribe of Northern California as they counted the spaces between fingers 10 (decimal) 12 (duodecimal) – dozen, gross, 24 hours in a day 20 (vigisimal) – Mayan, central & western Africa 60 (sexagesimal) – Sumeria, Mesopotamia Activity: Chip Trading

Your Number System