Transcript Winter Whiteland

PEMSTL Midwinter 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, realworld 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