Logical Arguments and Proof

scaleemptyΗλεκτρονική - Συσκευές

10 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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Logical Arguments and Proof


Students develop the concept of formal
proof and the use of theorems, postulates
and definitions; and use reasoning to defend
their conclusions using precise mathematical
language and symbols.


Students will be able to distingu
ish between
inductive and deductive reasoning and use
each

to test conjectures
or

prove geometric
statements

are true
.


Students will be able to write the converse,
inverse, and contrapositive of a geometric
statement and determine the validity of each
s
tatement.


Ex:
All squares have four right angles.



If a figure has four right angles, then it is
a square. Valid or invalid?





Students will be able to identify errors or
gaps in mathematical arguments and
develop counterexamples to refute invalid
st
atements.









Two
-

and Three
-
Dimensional
Figures


Students know and can prove theorems
about 2
-

and 3
-

dimensional figures,
including congruence, similarity and other
properties. Triangles and quadrilaterals are
the primary focus, but they also extend

their
learning to other polygons, circles and 3
-

dimensional figures.


Students will be able to
know

and apply
postulates and theorems about triangles,
including properties of special right
triangles,
to prove triangle congruence,
similarity and other pro
perties of triangles.


Ex:
If one leg of a right triangle has length
5 units and the adjacent angle is 30°,
determine the length of the other leg and the
hypotenuse.


Students will be able to solve problems
involving right triangles using the
Pythagorean
Theorem and trigonometric
ratios.


Students will be able to prove and apply
theorems about parallelograms and other
quadrilaterals and polygons, in addition to
working with circles and lines relating to
circles.






Geometry in the Coordinate Plane


Stude
nts make connections between
Geometry and Algebra by using the
coordinate plane to solve problems and
represent situations that are both purely
mathematical and that arise in applied
contexts.


Students will be able to determine the
equation of a line that

is described
geometrically and determine the coordinates
of a point described geometrically in the
coordinate plane.


Ex:
Write an equation of the line for a
perpendicular bisector of a given line
segment.


Ex:
Determine the coordinates of the
midpoint
of a given line segment.


Students will be able to verify and apply
properties of triangles and quadrilaterals in
the coordinate plane.


Ex:
Given a parallelogram on a coordinate
plane, verify that the diagonals bisect each
other.


Students will be able t
o write the equation of
a circle that is described geometrically in the
coordinate plane.






Lines and Angles


Students extend their understanding of
properties of parallel and perpendicular
lines, proving and applying theorems to
solve mathematical and
practical problems.


Students will be able to prove and apply
theorems about parallel and perpendicular
lines and theorems about angles.


Ex:

Prove that a point on the perpendicular
bisector of a line

segment

is equidistant
from the endpoints of the line

segment.


Students will be able to perform basic
constructions related to parallel and
perpendicular lines.


Geometric Transformations


Students deepen their understanding of
transformations, exploring and applying
properties of transformations.


Students

will be able to describe and sketch
transformations or composites of
transformations on a coordinate plane and
describe a rule for performing translations,
reflections across a line or rotations about a
point.






Additional Key Content


Students extend
and formalize their work
with geometric formulas for perimeter, area,
surface area and volume of two
-

and three
-
dimensional figures focusing on
mathematical derivations of these formulas
and the applications in complex problems.


Students will be able to d
erive and apply
formulas for arc length and area of a sector
of a circle.


Students will be able to apply formulas for
surface area and volume of three
-
dimensional figures and predict and verify
the effect
that

changing one, two or three
linear dimensions
will have on the surface
area or volume.


Ex:

What happens to the volume of a
cylinder if the length of the radius of the
base is doubled?


Students will be able to use and justify using
different degrees of precision to obtain
measurements.


Students wil
l be able to solve problems and
analyze results involving measurement
conversions within and between systems.


Ex:

A car averaged 38 miles per gallon on a
road trip. Determine the kilometers per liter
for this car.





____________School District’s


Pare
nt Guide

To


Washington State

Mathematics Standards



Geometry







LOGO








Additional Resources:




Textbook Website


District Website


Washington State OSPI

www.k12.wa.us